| L(s) = 1 | − 32·2-s − 108·3-s + 640·4-s − 12·5-s + 3.45e3·6-s − 1.37e3·7-s − 1.02e4·8-s + 7.29e3·9-s + 384·10-s + 5.32e3·11-s − 6.91e4·12-s − 4.03e3·13-s + 4.39e4·14-s + 1.29e3·15-s + 1.43e5·16-s − 2.53e4·17-s − 2.33e5·18-s − 9.53e3·19-s − 7.68e3·20-s + 1.48e5·21-s − 1.70e5·22-s + 4.25e4·23-s + 1.10e6·24-s − 1.55e5·25-s + 1.28e5·26-s − 3.93e5·27-s − 8.78e5·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s − 0.0429·5-s + 6.53·6-s − 1.51·7-s − 7.07·8-s + 10/3·9-s + 0.121·10-s + 1.20·11-s − 11.5·12-s − 0.508·13-s + 4.27·14-s + 0.0991·15-s + 35/4·16-s − 1.24·17-s − 9.42·18-s − 0.319·19-s − 0.214·20-s + 3.49·21-s − 3.41·22-s + 0.729·23-s + 16.3·24-s − 1.98·25-s + 1.43·26-s − 3.84·27-s − 7.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 11 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 12 T + 155469 T^{2} + 2448906 T^{3} + 3124258552 p T^{4} + 2448906 p^{7} T^{5} + 155469 p^{14} T^{6} + 12 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 310 p T + 152240557 T^{2} + 242498656186 T^{3} + 11404826186664316 T^{4} + 242498656186 p^{7} T^{5} + 152240557 p^{14} T^{6} + 310 p^{22} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 25304 T + 1121566812 T^{2} + 22819311347040 T^{3} + 571025555613465430 T^{4} + 22819311347040 p^{7} T^{5} + 1121566812 p^{14} T^{6} + 25304 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 502 p T + 2472742379 T^{2} + 25282716823284 T^{3} + 2884022088333754520 T^{4} + 25282716823284 p^{7} T^{5} + 2472742379 p^{14} T^{6} + 502 p^{22} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 42546 T + 3713424000 T^{2} - 57138082504314 T^{3} + 15121987949024026142 T^{4} - 57138082504314 p^{7} T^{5} + 3713424000 p^{14} T^{6} - 42546 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 19610 T + 55082996861 T^{2} + 74369583096470 T^{3} + \)\(12\!\cdots\!36\)\( T^{4} + 74369583096470 p^{7} T^{5} + 55082996861 p^{14} T^{6} + 19610 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 71568 T + 59823499148 T^{2} - 6683444203793832 T^{3} + \)\(19\!\cdots\!22\)\( T^{4} - 6683444203793832 p^{7} T^{5} + 59823499148 p^{14} T^{6} - 71568 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 200518 T + 169985739257 T^{2} + 52608623727733074 T^{3} + \)\(17\!\cdots\!48\)\( T^{4} + 52608623727733074 p^{7} T^{5} + 169985739257 p^{14} T^{6} + 200518 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 364010 T + 776900101620 T^{2} - 207727155214233678 T^{3} + \)\(22\!\cdots\!54\)\( T^{4} - 207727155214233678 p^{7} T^{5} + 776900101620 p^{14} T^{6} - 364010 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 47238 T + 921746754680 T^{2} + 57172280483041302 T^{3} + \)\(35\!\cdots\!86\)\( T^{4} + 57172280483041302 p^{7} T^{5} + 921746754680 p^{14} T^{6} + 47238 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 530470 T + 2107179697091 T^{2} + 807051160988063380 T^{3} + \)\(16\!\cdots\!28\)\( T^{4} + 807051160988063380 p^{7} T^{5} + 2107179697091 p^{14} T^{6} + 530470 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 479494 T + 921822258900 T^{2} - 672495286236420498 T^{3} - \)\(41\!\cdots\!82\)\( T^{4} - 672495286236420498 p^{7} T^{5} + 921822258900 p^{14} T^{6} - 479494 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1062220 T + 10157965259751 T^{2} - 7863800574650066040 T^{3} + \)\(38\!\cdots\!16\)\( T^{4} - 7863800574650066040 p^{7} T^{5} + 10157965259751 p^{14} T^{6} - 1062220 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 606532 T + 10299477812740 T^{2} - 5043210193008806956 T^{3} + \)\(45\!\cdots\!54\)\( T^{4} - 5043210193008806956 p^{7} T^{5} + 10299477812740 p^{14} T^{6} - 606532 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 3968348 T + 13844926908227 T^{2} + 40634262761587801044 T^{3} + \)\(13\!\cdots\!68\)\( T^{4} + 40634262761587801044 p^{7} T^{5} + 13844926908227 p^{14} T^{6} + 3968348 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 479464 T + 11725513064444 T^{2} + 109198252414341640 p T^{3} + \)\(66\!\cdots\!98\)\( T^{4} + 109198252414341640 p^{8} T^{5} + 11725513064444 p^{14} T^{6} - 479464 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 3785008 T + 13145584282669 T^{2} + 35524151594393597894 T^{3} - \)\(12\!\cdots\!92\)\( T^{4} + 35524151594393597894 p^{7} T^{5} + 13145584282669 p^{14} T^{6} - 3785008 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 7379372 T + 88640873413580 T^{2} - \)\(42\!\cdots\!24\)\( T^{3} + \)\(26\!\cdots\!78\)\( T^{4} - \)\(42\!\cdots\!24\)\( p^{7} T^{5} + 88640873413580 p^{14} T^{6} - 7379372 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 13997680 T + 138978052640700 T^{2} - \)\(88\!\cdots\!36\)\( T^{3} + \)\(51\!\cdots\!30\)\( T^{4} - \)\(88\!\cdots\!36\)\( p^{7} T^{5} + 138978052640700 p^{14} T^{6} - 13997680 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 5347948 T + 86569748459364 T^{2} - \)\(43\!\cdots\!64\)\( T^{3} + \)\(53\!\cdots\!70\)\( T^{4} - \)\(43\!\cdots\!64\)\( p^{7} T^{5} + 86569748459364 p^{14} T^{6} - 5347948 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 17834174 T + 340599244521188 T^{2} - 37763164341122026482 p T^{3} + \)\(40\!\cdots\!90\)\( T^{4} - 37763164341122026482 p^{8} T^{5} + 340599244521188 p^{14} T^{6} - 17834174 p^{21} T^{7} + p^{28} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.45319436803823514344003335776, −6.90938730622679442467114024160, −6.71639869980314011913988343879, −6.60169961207431897738372373289, −6.60123501143054613609364282804, −6.09699230944562962478995577959, −5.95729537893262274054871208753, −5.88007487872693016001191610499, −5.76983987038079384909453350987, −5.05331879634501889110469612074, −4.85156679602241923223619724482, −4.56718158280849364868056016779, −4.56689233216356107614317760624, −3.65312944426446363312718250152, −3.65102421691704682159459130557, −3.45972803957124483255718270855, −3.36263978714757235352866831969, −2.32509206752792879388764079413, −2.22551811950547136161841967676, −2.16639723039358972780184098805, −2.12918051684492677587684844678, −1.21671392728141086151721601452, −1.11799758114953976151249303958, −0.948487017649707406796638602192, −0.799406473074507554494604439583, 0, 0, 0, 0,
0.799406473074507554494604439583, 0.948487017649707406796638602192, 1.11799758114953976151249303958, 1.21671392728141086151721601452, 2.12918051684492677587684844678, 2.16639723039358972780184098805, 2.22551811950547136161841967676, 2.32509206752792879388764079413, 3.36263978714757235352866831969, 3.45972803957124483255718270855, 3.65102421691704682159459130557, 3.65312944426446363312718250152, 4.56689233216356107614317760624, 4.56718158280849364868056016779, 4.85156679602241923223619724482, 5.05331879634501889110469612074, 5.76983987038079384909453350987, 5.88007487872693016001191610499, 5.95729537893262274054871208753, 6.09699230944562962478995577959, 6.60123501143054613609364282804, 6.60169961207431897738372373289, 6.71639869980314011913988343879, 6.90938730622679442467114024160, 7.45319436803823514344003335776
