| L(s) = 1 | + 4·2-s + 12·4-s + 10·5-s − 2·7-s + 32·8-s + 40·10-s − 48·11-s − 80·13-s − 8·14-s + 80·16-s − 108·17-s − 188·19-s + 120·20-s − 192·22-s − 138·23-s + 75·25-s − 320·26-s − 24·28-s − 30·29-s − 188·31-s + 192·32-s − 432·34-s − 20·35-s − 332·37-s − 752·38-s + 320·40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.107·7-s + 1.41·8-s + 1.26·10-s − 1.31·11-s − 1.70·13-s − 0.152·14-s + 5/4·16-s − 1.54·17-s − 2.27·19-s + 1.34·20-s − 1.86·22-s − 1.25·23-s + 3/5·25-s − 2.41·26-s − 0.161·28-s − 0.192·29-s − 1.08·31-s + 1.06·32-s − 2.17·34-s − 0.0965·35-s − 1.47·37-s − 3.21·38-s + 1.26·40-s − 0.0228·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 393 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 48 T + 1702 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 5610 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 108 T + 12358 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 188 T + 18498 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 p T + 19009 T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 30 T + 42067 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 188 T + 55722 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 332 T + 97758 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 42595 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 556 T + 233898 T^{2} - 556 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 162 T - 4679 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 684 T + 414622 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 228 T + 178930 T^{2} - 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 254 T + 300747 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 826 T + 515001 T^{2} - 826 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 912 T + 729358 T^{2} + 912 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 232 T - 38814 T^{2} - 232 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 476 T + 480906 T^{2} + 476 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 306 T - 108143 T^{2} - 306 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1086 T + 1703923 T^{2} - 1086 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 868 T + 809478 T^{2} - 868 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−9.614043279780239568824442504743, −9.446224057386102636630186440230, −8.744107278065740252852514148161, −8.353708117010916593891338776116, −7.69477045468636157908344461999, −7.46636556415058898554561302479, −6.78665597283183610446397614783, −6.50457515712120085604895074899, −6.09324282429208822960184648318, −5.57393855982341921134269067257, −5.04582763795030504310360513612, −4.86548974278178968424799304494, −4.13737286118635816968480663026, −3.95355067386488660656451634928, −2.86940576472644504904604898230, −2.55510394254977837758909308098, −2.02683821566991618359215246420, −1.82929336534241510236328010650, 0, 0,
1.82929336534241510236328010650, 2.02683821566991618359215246420, 2.55510394254977837758909308098, 2.86940576472644504904604898230, 3.95355067386488660656451634928, 4.13737286118635816968480663026, 4.86548974278178968424799304494, 5.04582763795030504310360513612, 5.57393855982341921134269067257, 6.09324282429208822960184648318, 6.50457515712120085604895074899, 6.78665597283183610446397614783, 7.46636556415058898554561302479, 7.69477045468636157908344461999, 8.353708117010916593891338776116, 8.744107278065740252852514148161, 9.446224057386102636630186440230, 9.614043279780239568824442504743
