| L(s) = 1 | + 2-s − 8·3-s − 2·4-s − 25·5-s − 8·6-s + 6·8-s + 5·9-s − 25·10-s + 47·11-s + 16·12-s + 13-s + 200·15-s − 17·16-s − 2·17-s + 5·18-s − 21·19-s + 50·20-s + 47·22-s + 201·23-s − 48·24-s + 375·25-s + 26-s + 30·27-s + 190·29-s + 200·30-s + 388·31-s − 113·32-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.53·3-s − 1/4·4-s − 2.23·5-s − 0.544·6-s + 0.265·8-s + 5/27·9-s − 0.790·10-s + 1.28·11-s + 0.384·12-s + 0.0213·13-s + 3.44·15-s − 0.265·16-s − 0.0285·17-s + 0.0654·18-s − 0.253·19-s + 0.559·20-s + 0.455·22-s + 1.82·23-s − 0.408·24-s + 3·25-s + 0.00754·26-s + 0.213·27-s + 1.21·29-s + 1.21·30-s + 2.24·31-s − 0.624·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.717106047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717106047\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + p T )^{5} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 - T + 3 T^{2} - 11 T^{3} + 5 p^{3} T^{4} + p^{4} T^{5} + 5 p^{6} T^{6} - 11 p^{6} T^{7} + 3 p^{9} T^{8} - p^{12} T^{9} + p^{15} T^{10} \) |
| 3 | $C_2 \wr S_5$ | \( 1 + 8 T + 59 T^{2} + 134 p T^{3} + 955 p T^{4} + 14170 T^{5} + 955 p^{4} T^{6} + 134 p^{7} T^{7} + 59 p^{9} T^{8} + 8 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 47 T + 409 p T^{2} - 140260 T^{3} + 8055918 T^{4} - 207867546 T^{5} + 8055918 p^{3} T^{6} - 140260 p^{6} T^{7} + 409 p^{10} T^{8} - 47 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - T + 3333 T^{2} - 85816 T^{3} + 12357966 T^{4} - 60104190 T^{5} + 12357966 p^{3} T^{6} - 85816 p^{6} T^{7} + 3333 p^{9} T^{8} - p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + 2 T + 14265 T^{2} + 83744 T^{3} + 6177750 p T^{4} + 715660860 T^{5} + 6177750 p^{4} T^{6} + 83744 p^{6} T^{7} + 14265 p^{9} T^{8} + 2 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 21 T + 20535 T^{2} + 685376 T^{3} + 239855386 T^{4} + 5799664422 T^{5} + 239855386 p^{3} T^{6} + 685376 p^{6} T^{7} + 20535 p^{9} T^{8} + 21 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 201 T + 48989 T^{2} - 6224422 T^{3} + 1025553937 T^{4} - 103071638287 T^{5} + 1025553937 p^{3} T^{6} - 6224422 p^{6} T^{7} + 48989 p^{9} T^{8} - 201 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 190 T + 63587 T^{2} - 10646688 T^{3} + 1713461669 T^{4} - 306482100734 T^{5} + 1713461669 p^{3} T^{6} - 10646688 p^{6} T^{7} + 63587 p^{9} T^{8} - 190 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 388 T + 172703 T^{2} - 43786000 T^{3} + 10829267902 T^{4} - 1922674095576 T^{5} + 10829267902 p^{3} T^{6} - 43786000 p^{6} T^{7} + 172703 p^{9} T^{8} - 388 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 145 T + 136681 T^{2} + 19517508 T^{3} + 10851224194 T^{4} + 1121966157238 T^{5} + 10851224194 p^{3} T^{6} + 19517508 p^{6} T^{7} + 136681 p^{9} T^{8} + 145 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 281 T + 269231 T^{2} + 52376686 T^{3} + 31133332221 T^{4} + 4627148668935 T^{5} + 31133332221 p^{3} T^{6} + 52376686 p^{6} T^{7} + 269231 p^{9} T^{8} + 281 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 568 T + 328987 T^{2} - 103968566 T^{3} + 38129237297 T^{4} - 9424317397102 T^{5} + 38129237297 p^{3} T^{6} - 103968566 p^{6} T^{7} + 328987 p^{9} T^{8} - 568 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 + 473 T + 422915 T^{2} + 129059756 T^{3} + 68645838882 T^{4} + 16224661808822 T^{5} + 68645838882 p^{3} T^{6} + 129059756 p^{6} T^{7} + 422915 p^{9} T^{8} + 473 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 351 T + 708181 T^{2} - 191754208 T^{3} + 204450779430 T^{4} - 41520543155346 T^{5} + 204450779430 p^{3} T^{6} - 191754208 p^{6} T^{7} + 708181 p^{9} T^{8} - 351 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 12 p T + 1096227 T^{2} - 561741872 T^{3} + 464924363086 T^{4} - 171021556758360 T^{5} + 464924363086 p^{3} T^{6} - 561741872 p^{6} T^{7} + 1096227 p^{9} T^{8} - 12 p^{13} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 1944 T + 2577179 T^{2} - 2272501996 T^{3} + 1581534536149 T^{4} - 836610680685668 T^{5} + 1581534536149 p^{3} T^{6} - 2272501996 p^{6} T^{7} + 2577179 p^{9} T^{8} - 1944 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 1118 T + 1339679 T^{2} - 1021242718 T^{3} + 748444613473 T^{4} - 429178132979164 T^{5} + 748444613473 p^{3} T^{6} - 1021242718 p^{6} T^{7} + 1339679 p^{9} T^{8} - 1118 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 864 T + 1690919 T^{2} - 959528208 T^{3} + 1093514996270 T^{4} - 459054202031136 T^{5} + 1093514996270 p^{3} T^{6} - 959528208 p^{6} T^{7} + 1690919 p^{9} T^{8} - 864 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 1652 T + 2542033 T^{2} + 2388160624 T^{3} + 2082687842054 T^{4} + 1341889113242744 T^{5} + 2082687842054 p^{3} T^{6} + 2388160624 p^{6} T^{7} + 2542033 p^{9} T^{8} + 1652 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 218 T + 1501191 T^{2} - 164611504 T^{3} + 1132361617846 T^{4} - 88901695268332 T^{5} + 1132361617846 p^{3} T^{6} - 164611504 p^{6} T^{7} + 1501191 p^{9} T^{8} - 218 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 + 1502 T + 2544639 T^{2} + 2916185966 T^{3} + 2925083074177 T^{4} + 2334912952153996 T^{5} + 2925083074177 p^{3} T^{6} + 2916185966 p^{6} T^{7} + 2544639 p^{9} T^{8} + 1502 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 2322 T + 4506131 T^{2} - 5245245480 T^{3} + 5789036849341 T^{4} - 4794661589543922 T^{5} + 5789036849341 p^{3} T^{6} - 5245245480 p^{6} T^{7} + 4506131 p^{9} T^{8} - 2322 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 598 T + 4425037 T^{2} - 2054168072 T^{3} + 7929732195026 T^{4} - 2752469564393540 T^{5} + 7929732195026 p^{3} T^{6} - 2054168072 p^{6} T^{7} + 4425037 p^{9} T^{8} - 598 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.94454311036232921194851144966, −6.79878977114709442430582166535, −6.56049657469885735673878382480, −6.49602010793319599771784426256, −6.30408163407493739120109002436, −5.96505568831798632492283260067, −5.42854133290273831160391057278, −5.35117937631344771954764235488, −5.27261652133732265511669121279, −5.17262302880242219585273802202, −4.52324531184787193836306656766, −4.43303135347290612964298406750, −4.41792883063734429864965577134, −4.02974833085977721617996502202, −3.90745522485810928723563313172, −3.44814758247506059171419828911, −3.27095166916644933919952310672, −2.97707109187121412549108112270, −2.72753269079475876689169459759, −2.03048246197587978143469051478, −2.00719706056377392986009716192, −0.969431333372840012431431164011, −0.948381794672485627324477650540, −0.67130409657206205598495991732, −0.34562167186713641533082145738,
0.34562167186713641533082145738, 0.67130409657206205598495991732, 0.948381794672485627324477650540, 0.969431333372840012431431164011, 2.00719706056377392986009716192, 2.03048246197587978143469051478, 2.72753269079475876689169459759, 2.97707109187121412549108112270, 3.27095166916644933919952310672, 3.44814758247506059171419828911, 3.90745522485810928723563313172, 4.02974833085977721617996502202, 4.41792883063734429864965577134, 4.43303135347290612964298406750, 4.52324531184787193836306656766, 5.17262302880242219585273802202, 5.27261652133732265511669121279, 5.35117937631344771954764235488, 5.42854133290273831160391057278, 5.96505568831798632492283260067, 6.30408163407493739120109002436, 6.49602010793319599771784426256, 6.56049657469885735673878382480, 6.79878977114709442430582166535, 6.94454311036232921194851144966
