| L(s) = 1 | + 20·3-s + 250·5-s − 1.66e3·7-s + 442·9-s − 3.60e3·11-s + 1.31e4·13-s + 5.00e3·15-s + 5.46e3·17-s + 4.04e4·19-s − 3.32e4·21-s + 4.18e4·23-s + 4.68e4·25-s + 5.34e4·27-s + 1.18e5·29-s + 1.15e5·31-s − 7.20e4·33-s − 4.15e5·35-s + 3.06e5·37-s + 2.63e5·39-s − 3.53e5·41-s − 1.21e6·43-s + 1.10e5·45-s + 2.06e6·47-s + 7.85e5·49-s + 1.09e5·51-s + 1.40e6·53-s − 9.00e5·55-s + ⋯ |
| L(s) = 1 | + 0.427·3-s + 0.894·5-s − 1.82·7-s + 0.202·9-s − 0.815·11-s + 1.66·13-s + 0.382·15-s + 0.269·17-s + 1.35·19-s − 0.782·21-s + 0.716·23-s + 3/5·25-s + 0.522·27-s + 0.903·29-s + 0.698·31-s − 0.348·33-s − 1.63·35-s + 0.996·37-s + 0.711·39-s − 0.800·41-s − 2.33·43-s + 0.180·45-s + 2.90·47-s + 0.953·49-s + 0.115·51-s + 1.29·53-s − 0.729·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.498190459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.498190459\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 20 T - 14 p T^{2} - 20 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 1660 T + 1970190 T^{2} + 1660 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3600 T + 5634742 T^{2} + 3600 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 13180 T + 163072398 T^{2} - 13180 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5460 T - 160982138 T^{2} - 5460 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 40472 T + 2050920774 T^{2} - 40472 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 41820 T + 7228954990 T^{2} - 41820 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4092 p T + 37435002574 T^{2} - 4092 p^{8} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 115928 T + 13575043518 T^{2} - 115928 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 306940 T + 120498758430 T^{2} - 306940 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 353148 T + 314020811638 T^{2} + 353148 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1215340 T + 866800610214 T^{2} + 1215340 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2068500 T + 2082813588382 T^{2} - 2068500 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1400460 T + 2025459025630 T^{2} - 1400460 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1992504 T + 5477605919542 T^{2} - 1992504 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1678676 T + 900787415886 T^{2} + 1678676 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3663940 T + 12525094986870 T^{2} + 3663940 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1794936 T + 17428289847406 T^{2} + 1794936 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5062180 T + 19516555685238 T^{2} - 5062180 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10178224 T + 57668559431262 T^{2} + 10178224 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7214100 T + 67277011758070 T^{2} - 7214100 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15330828 T + 138366637288054 T^{2} + 15330828 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14024020 T + 210744479822310 T^{2} - 14024020 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.27253681411043350365833099794, −13.00463275148248776283389069335, −12.23406625088310727288236885091, −11.70940655381887916635946717853, −10.81893892811333185126520040889, −10.25173004700758752712961097381, −9.941251905819841219543396041361, −9.414519060959878371516239589161, −8.690250815172912725659968299462, −8.378123818858546491364324736349, −7.26381462238656931876866040475, −6.83562664867804186651113832998, −6.04949812225145502990323633587, −5.74007124027704651413264820573, −4.80769715225556180091069654398, −3.69626138462585054835574241027, −3.05364650917635471686614839267, −2.66699085073468423814450120089, −1.34359537403694219737659970990, −0.66286536679789238902129535725,
0.66286536679789238902129535725, 1.34359537403694219737659970990, 2.66699085073468423814450120089, 3.05364650917635471686614839267, 3.69626138462585054835574241027, 4.80769715225556180091069654398, 5.74007124027704651413264820573, 6.04949812225145502990323633587, 6.83562664867804186651113832998, 7.26381462238656931876866040475, 8.378123818858546491364324736349, 8.690250815172912725659968299462, 9.414519060959878371516239589161, 9.941251905819841219543396041361, 10.25173004700758752712961097381, 10.81893892811333185126520040889, 11.70940655381887916635946717853, 12.23406625088310727288236885091, 13.00463275148248776283389069335, 13.27253681411043350365833099794
