| L(s) = 1 | − 4·2-s + 12·4-s − 5·5-s − 32·8-s + 20·10-s − 67·11-s + 41·13-s + 80·16-s + 92·17-s + 43·19-s − 60·20-s + 268·22-s − 148·23-s + 105·25-s − 164·26-s − 77·29-s − 520·31-s − 192·32-s − 368·34-s + 7·37-s − 172·38-s + 160·40-s + 426·41-s − 107·43-s − 804·44-s + 592·46-s − 576·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.447·5-s − 1.41·8-s + 0.632·10-s − 1.83·11-s + 0.874·13-s + 5/4·16-s + 1.31·17-s + 0.519·19-s − 0.670·20-s + 2.59·22-s − 1.34·23-s + 0.839·25-s − 1.23·26-s − 0.493·29-s − 3.01·31-s − 1.06·32-s − 1.85·34-s + 0.0311·37-s − 0.734·38-s + 0.632·40-s + 1.62·41-s − 0.379·43-s − 2.75·44-s + 1.89·46-s − 1.78·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + p T - 16 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 67 T + 3448 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 41 T + 4478 T^{2} - 41 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 92 T + 386 p T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 43 T + 11154 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 148 T + 24430 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 77 T + 9574 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 520 T + 125837 T^{2} + 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 74082 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 426 T + 171106 T^{2} - 426 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 107 T + 86220 T^{2} + 107 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 576 T + 242170 T^{2} + 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 243 T + 309490 T^{2} - 243 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 200614 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 224 T + 461126 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 687 T + 678832 T^{2} - 687 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 472 T + 637018 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 921 T + 578188 T^{2} + 921 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 526 T + 1033727 T^{2} + 526 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 221 T + 945628 T^{2} - 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 774 T + 966562 T^{2} + 774 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1953 T + 2366992 T^{2} - 1953 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.479341125775286666430802948915, −9.202628686070718957885959264752, −8.590108869523462878104662957699, −8.287278464535227622634184123902, −7.73918532983005131911499699871, −7.73532615399993554119077673714, −7.20098743960506249640390029449, −6.81033209457823173796239785224, −5.97016851032345280514757729007, −5.69221429673130944695147216416, −5.38867215551372079608824636540, −4.73926400747005644746231423655, −3.71279319757000889726392222899, −3.62173075083196321389830684556, −2.85697776883603054818249490753, −2.34800380461761818977615785948, −1.61008240288151276559578093907, −1.08032784975972696254875777730, 0, 0,
1.08032784975972696254875777730, 1.61008240288151276559578093907, 2.34800380461761818977615785948, 2.85697776883603054818249490753, 3.62173075083196321389830684556, 3.71279319757000889726392222899, 4.73926400747005644746231423655, 5.38867215551372079608824636540, 5.69221429673130944695147216416, 5.97016851032345280514757729007, 6.81033209457823173796239785224, 7.20098743960506249640390029449, 7.73532615399993554119077673714, 7.73918532983005131911499699871, 8.287278464535227622634184123902, 8.590108869523462878104662957699, 9.202628686070718957885959264752, 9.479341125775286666430802948915
