L(s) = 1 | + 2·3-s + 2·5-s + 2·9-s + 14·11-s + 14·13-s + 4·15-s + 2·19-s − 22·23-s + 8·25-s − 2·27-s − 6·31-s + 28·33-s + 28·39-s − 8·41-s − 14·43-s + 4·45-s + 16·49-s − 8·53-s + 28·55-s + 4·57-s + 2·59-s − 12·61-s + 28·65-s + 20·67-s − 44·69-s − 22·71-s − 28·73-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.894·5-s + 2/3·9-s + 4.22·11-s + 3.88·13-s + 1.03·15-s + 0.458·19-s − 4.58·23-s + 8/5·25-s − 0.384·27-s − 1.07·31-s + 4.87·33-s + 4.48·39-s − 1.24·41-s − 2.13·43-s + 0.596·45-s + 16/7·49-s − 1.09·53-s + 3.77·55-s + 0.529·57-s + 0.260·59-s − 1.53·61-s + 3.47·65-s + 2.44·67-s − 5.29·69-s − 2.61·71-s − 3.27·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.087535531\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.087535531\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 2 T - 4 T^{2} + 2 p T^{3} - 2 T^{4} + 2 p^{2} T^{5} - 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | \( 1 - 14 T + 112 T^{2} - 618 T^{3} + 2550 T^{4} - 618 p T^{5} + 112 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
good | 3 | \( 1 - 2 T + 2 T^{2} + 2 T^{3} - 4 T^{4} - 2 T^{5} + 14 T^{6} + 26 T^{7} - 86 T^{8} + 26 p T^{9} + 14 p^{2} T^{10} - 2 p^{3} T^{11} - 4 p^{4} T^{12} + 2 p^{5} T^{13} + 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 16 T^{2} + 148 T^{4} - 736 T^{6} + 3910 T^{8} - 736 p^{2} T^{10} + 148 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 14 T + 98 T^{2} - 486 T^{3} + 2120 T^{4} - 9034 T^{5} + 36814 T^{6} - 138306 T^{7} + 477778 T^{8} - 138306 p T^{9} + 36814 p^{2} T^{10} - 9034 p^{3} T^{11} + 2120 p^{4} T^{12} - 486 p^{5} T^{13} + 98 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( 1 + 8 T^{3} - 116 T^{4} - 216 T^{5} + 32 T^{6} - 2224 T^{7} + 100710 T^{8} - 2224 p T^{9} + 32 p^{2} T^{10} - 216 p^{3} T^{11} - 116 p^{4} T^{12} + 8 p^{5} T^{13} + p^{8} T^{16} \) |
| 19 | \( 1 - 2 T + 2 T^{2} + 10 T^{3} - 584 T^{4} + 294 T^{5} + 630 T^{6} - 14718 T^{7} + 297906 T^{8} - 14718 p T^{9} + 630 p^{2} T^{10} + 294 p^{3} T^{11} - 584 p^{4} T^{12} + 10 p^{5} T^{13} + 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 + 22 T + 242 T^{2} + 2022 T^{3} + 15212 T^{4} + 101978 T^{5} + 606454 T^{6} + 3329826 T^{7} + 16823818 T^{8} + 3329826 p T^{9} + 606454 p^{2} T^{10} + 101978 p^{3} T^{11} + 15212 p^{4} T^{12} + 2022 p^{5} T^{13} + 242 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 - 76 T^{2} + 4540 T^{4} - 184020 T^{6} + 6139446 T^{8} - 184020 p^{2} T^{10} + 4540 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( 1 + 6 T + 18 T^{2} + 142 T^{3} + 1144 T^{4} + 5034 T^{5} + 19694 T^{6} + 210370 T^{7} + 2234226 T^{8} + 210370 p T^{9} + 19694 p^{2} T^{10} + 5034 p^{3} T^{11} + 1144 p^{4} T^{12} + 142 p^{5} T^{13} + 18 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 220 T^{2} + 23308 T^{4} - 1534084 T^{6} + 68279254 T^{8} - 1534084 p^{2} T^{10} + 23308 p^{4} T^{12} - 220 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 + 8 T + 32 T^{2} + 176 T^{3} + 2172 T^{4} + 16928 T^{5} + 81408 T^{6} + 405608 T^{7} + 1063750 T^{8} + 405608 p T^{9} + 81408 p^{2} T^{10} + 16928 p^{3} T^{11} + 2172 p^{4} T^{12} + 176 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 + 14 T + 98 T^{2} + 10 p T^{3} - 2020 T^{4} - 30806 T^{5} - 140874 T^{6} - 92190 T^{7} + 3625258 T^{8} - 92190 p T^{9} - 140874 p^{2} T^{10} - 30806 p^{3} T^{11} - 2020 p^{4} T^{12} + 10 p^{6} T^{13} + 98 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 + 24 T^{2} + 2308 T^{4} + 38424 T^{6} + 1085670 T^{8} + 38424 p^{2} T^{10} + 2308 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 8 T + 32 T^{2} - 504 T^{3} + 476 T^{4} + 50968 T^{5} + 519520 T^{6} + 102360 T^{7} - 12998234 T^{8} + 102360 p T^{9} + 519520 p^{2} T^{10} + 50968 p^{3} T^{11} + 476 p^{4} T^{12} - 504 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 - 2 T + 2 T^{2} - 70 T^{3} + 104 p T^{4} - 12666 T^{5} + 15510 T^{6} - 711918 T^{7} + 32447346 T^{8} - 711918 p T^{9} + 15510 p^{2} T^{10} - 12666 p^{3} T^{11} + 104 p^{5} T^{12} - 70 p^{5} T^{13} + 2 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 6 T + 76 T^{2} - 278 T^{3} + 378 T^{4} - 278 p T^{5} + 76 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 10 T + 4 p T^{2} - 1986 T^{3} + 26922 T^{4} - 1986 p T^{5} + 4 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 22 T + 242 T^{2} + 2058 T^{3} + 14096 T^{4} + 3350 T^{5} - 1219850 T^{6} - 18442086 T^{7} - 179382110 T^{8} - 18442086 p T^{9} - 1219850 p^{2} T^{10} + 3350 p^{3} T^{11} + 14096 p^{4} T^{12} + 2058 p^{5} T^{13} + 242 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 14 T + 124 T^{2} - 318 T^{3} - 5766 T^{4} - 318 p T^{5} + 124 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 292 T^{2} + 39172 T^{4} - 3359068 T^{6} + 252263350 T^{8} - 3359068 p^{2} T^{10} + 39172 p^{4} T^{12} - 292 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 504 T^{2} + 117748 T^{4} - 16965208 T^{6} + 1675348902 T^{8} - 16965208 p^{2} T^{10} + 117748 p^{4} T^{12} - 504 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 4 T + 8 T^{2} - 204 T^{3} - 9364 T^{4} + 35164 T^{5} - 44936 T^{6} + 55860 T^{7} + 65266150 T^{8} + 55860 p T^{9} - 44936 p^{2} T^{10} + 35164 p^{3} T^{11} - 9364 p^{4} T^{12} - 204 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 6 T + 172 T^{2} + 446 T^{3} + 10158 T^{4} + 446 p T^{5} + 172 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.65040418567341855585470419539, −5.48909518327577227843558348305, −5.07036635060196351533026050046, −4.69644136395765018055023067431, −4.65951416293981164169589773253, −4.61191879187748414823018152045, −4.53114807776115897715715771494, −4.06172929195261134720903881447, −4.05633943027093866350377189028, −4.00029948847555936878921780999, −3.74334118231884518986449846817, −3.73926103348645964961103367730, −3.47270713777382595923307941713, −3.43854011658717198517641913230, −3.12289038825013155740295364735, −3.06267358520805638901662311957, −3.01048754699987422437820009950, −2.33531254163976951545207352373, −2.04862407886934408911503799930, −1.99177445028314580724706508339, −1.72616182262416600700108333457, −1.59446527235327441419065788759, −1.57310929837113236097200419167, −1.14228121568982728795594409616, −0.823465290794851858321158793568,
0.823465290794851858321158793568, 1.14228121568982728795594409616, 1.57310929837113236097200419167, 1.59446527235327441419065788759, 1.72616182262416600700108333457, 1.99177445028314580724706508339, 2.04862407886934408911503799930, 2.33531254163976951545207352373, 3.01048754699987422437820009950, 3.06267358520805638901662311957, 3.12289038825013155740295364735, 3.43854011658717198517641913230, 3.47270713777382595923307941713, 3.73926103348645964961103367730, 3.74334118231884518986449846817, 4.00029948847555936878921780999, 4.05633943027093866350377189028, 4.06172929195261134720903881447, 4.53114807776115897715715771494, 4.61191879187748414823018152045, 4.65951416293981164169589773253, 4.69644136395765018055023067431, 5.07036635060196351533026050046, 5.48909518327577227843558348305, 5.65040418567341855585470419539
Plot not available for L-functions of degree greater than 10.