| L(s) = 1 | + 2·2-s − 5·5-s − 32·7-s − 8·8-s − 10·10-s + 60·11-s + 34·13-s − 64·14-s − 16·16-s + 84·17-s − 152·19-s + 120·22-s + 68·26-s − 6·29-s + 232·31-s + 168·34-s + 160·35-s + 268·37-s − 304·38-s + 40·40-s − 234·41-s + 412·43-s + 360·47-s + 343·49-s + 444·53-s − 300·55-s + 256·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s − 1.72·7-s − 0.353·8-s − 0.316·10-s + 1.64·11-s + 0.725·13-s − 1.22·14-s − 1/4·16-s + 1.19·17-s − 1.83·19-s + 1.16·22-s + 0.512·26-s − 0.0384·29-s + 1.34·31-s + 0.847·34-s + 0.772·35-s + 1.19·37-s − 1.29·38-s + 0.158·40-s − 0.891·41-s + 1.46·43-s + 1.11·47-s + 49-s + 1.15·53-s − 0.735·55-s + 0.610·56-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
| \(L(2)\) |
\(\approx\) |
\(3.212996795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.212996795\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 32 T + 681 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 60 T + 2269 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 34 T - 1041 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 42 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T - 24353 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 232 T + 24033 T^{2} - 232 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 134 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 234 T - 14165 T^{2} + 234 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 412 T + 90237 T^{2} - 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 360 T + 25777 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 222 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 660 T + 230221 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 490 T + 13119 T^{2} - 490 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 812 T + 358581 T^{2} + 812 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 746 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 152 T - 469935 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 804 T + 74629 T^{2} - 804 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 678 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 p T - 93 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 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
−10.01911028492347273981096233158, −9.479096944857846219667136653748, −9.405671121307096273759102441707, −8.848214881277306948057079313164, −8.395208822307702350321096523819, −8.029740175808067916509861505565, −7.37802125234485328490807208166, −6.68369849093669125425275941762, −6.64842736603105063638981962837, −6.01368238490080781420393991091, −5.97891090054215909030413828864, −5.21393995428221737239292592381, −4.36608782396346041358153334149, −4.04265076488216819634517519659, −3.87400883850698906736110299304, −3.14765971809509804338832196463, −2.83060702468102597385727521336, −1.95693317656118713247152338428, −0.988237826964075403564759738576, −0.52284306338248460731448297078,
0.52284306338248460731448297078, 0.988237826964075403564759738576, 1.95693317656118713247152338428, 2.83060702468102597385727521336, 3.14765971809509804338832196463, 3.87400883850698906736110299304, 4.04265076488216819634517519659, 4.36608782396346041358153334149, 5.21393995428221737239292592381, 5.97891090054215909030413828864, 6.01368238490080781420393991091, 6.64842736603105063638981962837, 6.68369849093669125425275941762, 7.37802125234485328490807208166, 8.029740175808067916509861505565, 8.395208822307702350321096523819, 8.848214881277306948057079313164, 9.405671121307096273759102441707, 9.479096944857846219667136653748, 10.01911028492347273981096233158
