| L(s) = 1 | + 2·3-s − 2·4-s + 2·5-s + 3·9-s + 2·11-s − 4·12-s + 2·13-s + 4·15-s − 8·17-s − 4·19-s − 4·20-s − 2·23-s + 3·25-s + 4·27-s − 6·29-s − 4·31-s + 4·33-s − 6·36-s − 4·37-s + 4·39-s − 2·41-s + 2·43-s − 4·44-s + 6·45-s − 6·47-s − 16·51-s − 4·52-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 0.894·5-s + 9-s + 0.603·11-s − 1.15·12-s + 0.554·13-s + 1.03·15-s − 1.94·17-s − 0.917·19-s − 0.894·20-s − 0.417·23-s + 3/5·25-s + 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s − 36-s − 0.657·37-s + 0.640·39-s − 0.312·41-s + 0.304·43-s − 0.603·44-s + 0.894·45-s − 0.875·47-s − 2.24·51-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65367225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65367225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−7.64000092356264588166809454373, −7.32639307466582776516583438218, −7.03052757236544294331681061230, −6.51347085236681106658345614876, −6.34791061824034747649289960933, −6.09476505032441282593161878714, −5.37327955729289358680483433672, −5.29942185696381565580199266283, −4.67137571031139663458273618591, −4.32116337802406664881485568311, −4.13662063776252347515921238477, −3.85817434511139899792998173297, −3.25618347147221498142368689726, −2.92208550979857362036705179326, −2.38393805958349612054865042088, −1.97851571494229361553057113409, −1.63221905933814101247354238457, −1.28498711754768590141123245761, 0, 0,
1.28498711754768590141123245761, 1.63221905933814101247354238457, 1.97851571494229361553057113409, 2.38393805958349612054865042088, 2.92208550979857362036705179326, 3.25618347147221498142368689726, 3.85817434511139899792998173297, 4.13662063776252347515921238477, 4.32116337802406664881485568311, 4.67137571031139663458273618591, 5.29942185696381565580199266283, 5.37327955729289358680483433672, 6.09476505032441282593161878714, 6.34791061824034747649289960933, 6.51347085236681106658345614876, 7.03052757236544294331681061230, 7.32639307466582776516583438218, 7.64000092356264588166809454373
