| L(s) = 1 | + 5·2-s + 15·4-s − 4·5-s + 35·8-s − 7·9-s − 20·10-s + 4·11-s + 70·16-s + 14·17-s − 35·18-s − 10·19-s − 60·20-s + 20·22-s − 5·25-s + 2·27-s + 12·29-s − 5·31-s + 126·32-s + 70·34-s − 105·36-s − 50·38-s − 140·40-s + 2·41-s − 4·43-s + 60·44-s + 28·45-s + 40·47-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 15/2·4-s − 1.78·5-s + 12.3·8-s − 7/3·9-s − 6.32·10-s + 1.20·11-s + 35/2·16-s + 3.39·17-s − 8.24·18-s − 2.29·19-s − 13.4·20-s + 4.26·22-s − 25-s + 0.384·27-s + 2.22·29-s − 0.898·31-s + 22.2·32-s + 12.0·34-s − 17.5·36-s − 8.11·38-s − 22.1·40-s + 0.312·41-s − 0.609·43-s + 9.04·44-s + 4.17·45-s + 5.83·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 31^{5} \cdot 97^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 31^{5} \cdot 97^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(92.55060948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(92.55060948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{5} \) |
| 31 | $C_1$ | \( ( 1 + T )^{5} \) |
| 97 | $C_1$ | \( ( 1 + T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 + 7 T^{2} - 2 T^{3} + 23 T^{4} - 10 T^{5} + 23 p T^{6} - 2 p^{2} T^{7} + 7 p^{3} T^{8} + p^{5} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 + 4 T + 21 T^{2} + 2 p^{2} T^{3} + 163 T^{4} + 298 T^{5} + 163 p T^{6} + 2 p^{4} T^{7} + 21 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{5} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 4 T + 47 T^{2} - 14 p T^{3} + 959 T^{4} - 2446 T^{5} + 959 p T^{6} - 14 p^{3} T^{7} + 47 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 25 T^{2} + 32 T^{3} + 402 T^{4} + 704 T^{5} + 402 p T^{6} + 32 p^{2} T^{7} + 25 p^{3} T^{8} + p^{5} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 14 T + 139 T^{2} - 904 T^{3} + 4977 T^{4} - 21638 T^{5} + 4977 p T^{6} - 904 p^{2} T^{7} + 139 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 10 T + 115 T^{2} + 706 T^{3} + 4539 T^{4} + 19436 T^{5} + 4539 p T^{6} + 706 p^{2} T^{7} + 115 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 + 25 T^{2} - 12 T^{3} + 921 T^{4} + 116 T^{5} + 921 p T^{6} - 12 p^{2} T^{7} + 25 p^{3} T^{8} + p^{5} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 12 T + 129 T^{2} - 688 T^{3} + 4106 T^{4} - 16392 T^{5} + 4106 p T^{6} - 688 p^{2} T^{7} + 129 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{5} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 2 T + 37 T^{2} + 8 T^{3} + 2210 T^{4} + 2036 T^{5} + 2210 p T^{6} + 8 p^{2} T^{7} + 37 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 + 4 T + 207 T^{2} + 666 T^{3} + 17471 T^{4} + 42510 T^{5} + 17471 p T^{6} + 666 p^{2} T^{7} + 207 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 40 T + 797 T^{2} - 10576 T^{3} + 104729 T^{4} - 808656 T^{5} + 104729 p T^{6} - 10576 p^{2} T^{7} + 797 p^{3} T^{8} - 40 p^{4} T^{9} + p^{5} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 30 T + 553 T^{2} - 7170 T^{3} + 72343 T^{4} - 584676 T^{5} + 72343 p T^{6} - 7170 p^{2} T^{7} + 553 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 22 T + 387 T^{2} - 4530 T^{3} + 46739 T^{4} - 377552 T^{5} + 46739 p T^{6} - 4530 p^{2} T^{7} + 387 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 16 T + 361 T^{2} + 3808 T^{3} + 46870 T^{4} + 344992 T^{5} + 46870 p T^{6} + 3808 p^{2} T^{7} + 361 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 4 T + 167 T^{2} - 880 T^{3} + 18502 T^{4} - 65192 T^{5} + 18502 p T^{6} - 880 p^{2} T^{7} + 167 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 40 T + 955 T^{2} + 15552 T^{3} + 191794 T^{4} + 1821488 T^{5} + 191794 p T^{6} + 15552 p^{2} T^{7} + 955 p^{3} T^{8} + 40 p^{4} T^{9} + p^{5} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 18 T + 301 T^{2} - 2216 T^{3} + 18858 T^{4} - 89324 T^{5} + 18858 p T^{6} - 2216 p^{2} T^{7} + 301 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 20 T + 427 T^{2} - 5424 T^{3} + 67690 T^{4} - 606328 T^{5} + 67690 p T^{6} - 5424 p^{2} T^{7} + 427 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 38 T + 903 T^{2} - 14840 T^{3} + 189778 T^{4} - 1919492 T^{5} + 189778 p T^{6} - 14840 p^{2} T^{7} + 903 p^{3} T^{8} - 38 p^{4} T^{9} + p^{5} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 18 T + 389 T^{2} - 3736 T^{3} + 48658 T^{4} - 354540 T^{5} + 48658 p T^{6} - 3736 p^{2} T^{7} + 389 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \) |
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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
−4.54137476836578872561049849158, −4.53034583623858744489602534577, −4.38379895238542534624555658253, −4.28403936755104808656212217846, −4.18609049357142508794530395253, −3.99437184105129699067730598371, −3.71613478620329438785312443627, −3.65525047559225961795456040763, −3.54985895337392968033348276651, −3.47252487525466862171088181316, −3.23775592727945938445640138954, −3.15111318034888191476406416749, −2.92305110468265889702433725155, −2.84319259689133707543710568908, −2.69892567273867427878884866742, −2.23103576259638108442341575767, −2.06093187653422993498454145240, −2.03771872259451923744955125997, −1.95886143535481511359515352328, −1.89961695232508445556965562686, −1.18469475860936936299925716691, −0.930567867735705008052678466486, −0.817259984966020873462656897693, −0.63758180988956119582778755281, −0.41238170397150911278123351925,
0.41238170397150911278123351925, 0.63758180988956119582778755281, 0.817259984966020873462656897693, 0.930567867735705008052678466486, 1.18469475860936936299925716691, 1.89961695232508445556965562686, 1.95886143535481511359515352328, 2.03771872259451923744955125997, 2.06093187653422993498454145240, 2.23103576259638108442341575767, 2.69892567273867427878884866742, 2.84319259689133707543710568908, 2.92305110468265889702433725155, 3.15111318034888191476406416749, 3.23775592727945938445640138954, 3.47252487525466862171088181316, 3.54985895337392968033348276651, 3.65525047559225961795456040763, 3.71613478620329438785312443627, 3.99437184105129699067730598371, 4.18609049357142508794530395253, 4.28403936755104808656212217846, 4.38379895238542534624555658253, 4.53034583623858744489602534577, 4.54137476836578872561049849158
