| L(s) = 1 | − 14·5-s + 22·7-s − 6·11-s + 34·17-s − 38·19-s − 4·23-s + 97·25-s − 308·35-s − 42·43-s + 10·47-s + 265·49-s + 84·55-s + 46·61-s + 78·73-s − 132·77-s + 12·83-s − 476·85-s + 532·95-s − 244·101-s + 56·115-s + 748·119-s − 215·121-s − 322·125-s + 127-s + 131-s − 836·133-s + 137-s + ⋯ |
| L(s) = 1 | − 2.79·5-s + 22/7·7-s − 0.545·11-s + 2·17-s − 2·19-s − 0.173·23-s + 3.87·25-s − 8.79·35-s − 0.976·43-s + 0.212·47-s + 5.40·49-s + 1.52·55-s + 0.754·61-s + 1.06·73-s − 1.71·77-s + 0.144·83-s − 5.59·85-s + 28/5·95-s − 2.41·101-s + 0.486·115-s + 44/7·119-s − 1.77·121-s − 2.57·125-s + 0.00787·127-s + 0.00763·131-s − 6.28·133-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.571003742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.571003742\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 114 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1890 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1170 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 1794 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 21 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 5586 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7410 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1890 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 39 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3234 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1730 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )( 1 + 94 T + p^{2} T^{2} ) \) |
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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
−9.579471899502245359051133658681, −8.860014420994767024510751248918, −8.334390976379347207328400380183, −8.240758358445587314196312162055, −8.151532739750829528859180034958, −7.73429533160774630652618919763, −7.43061710651208913548008776354, −7.08593979365430745137786953972, −6.42119120180256845599308192195, −5.64280599633585729859919575834, −5.18798226344042811127633184941, −4.92343680501456391703904393159, −4.42479742106682676182510422101, −4.07577873662142852944064485427, −3.81297874915072946765816737683, −3.16968371537562519942490384499, −2.38986715855665952531074239790, −1.74298061695695176831058000434, −1.14005758116268903527423945457, −0.40844956299243013454826130448,
0.40844956299243013454826130448, 1.14005758116268903527423945457, 1.74298061695695176831058000434, 2.38986715855665952531074239790, 3.16968371537562519942490384499, 3.81297874915072946765816737683, 4.07577873662142852944064485427, 4.42479742106682676182510422101, 4.92343680501456391703904393159, 5.18798226344042811127633184941, 5.64280599633585729859919575834, 6.42119120180256845599308192195, 7.08593979365430745137786953972, 7.43061710651208913548008776354, 7.73429533160774630652618919763, 8.151532739750829528859180034958, 8.240758358445587314196312162055, 8.334390976379347207328400380183, 8.860014420994767024510751248918, 9.579471899502245359051133658681
