L(s) = 1 | + 1.68·2-s − 0.710·3-s + 0.833·4-s − 1.19·6-s − 4.59·7-s − 1.96·8-s − 2.49·9-s + 3.91·11-s − 0.592·12-s − 0.572·13-s − 7.72·14-s − 4.97·16-s − 0.232·17-s − 4.20·18-s − 5.55·19-s + 3.26·21-s + 6.59·22-s − 4.93·23-s + 1.39·24-s − 0.964·26-s + 3.90·27-s − 3.82·28-s + 4.13·29-s − 3.49·31-s − 4.44·32-s − 2.78·33-s − 0.391·34-s + ⋯ |
L(s) = 1 | + 1.19·2-s − 0.410·3-s + 0.416·4-s − 0.488·6-s − 1.73·7-s − 0.694·8-s − 0.831·9-s + 1.18·11-s − 0.170·12-s − 0.158·13-s − 2.06·14-s − 1.24·16-s − 0.0564·17-s − 0.990·18-s − 1.27·19-s + 0.711·21-s + 1.40·22-s − 1.02·23-s + 0.284·24-s − 0.189·26-s + 0.751·27-s − 0.723·28-s + 0.767·29-s − 0.627·31-s − 0.785·32-s − 0.484·33-s − 0.0671·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 - 1.68T + 2T^{2} \) |
| 3 | \( 1 + 0.710T + 3T^{2} \) |
| 7 | \( 1 + 4.59T + 7T^{2} \) |
| 11 | \( 1 - 3.91T + 11T^{2} \) |
| 13 | \( 1 + 0.572T + 13T^{2} \) |
| 17 | \( 1 + 0.232T + 17T^{2} \) |
| 19 | \( 1 + 5.55T + 19T^{2} \) |
| 23 | \( 1 + 4.93T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 + 3.49T + 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 1.38T + 43T^{2} \) |
| 47 | \( 1 - 0.920T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 - 3.20T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 9.61T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 7.25T + 89T^{2} \) |
| 97 | \( 1 + 8.31T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.25786420242732647206052933090, −9.272741101030207950672752381230, −8.668579440672815407748615791918, −6.96043039935193165794878074137, −6.17059024064099961578062859224, −5.84512846361201473729238192484, −4.40850571664778979311498514023, −3.62698187680388013616599577097, −2.63445469730603800009264649402, 0,
2.63445469730603800009264649402, 3.62698187680388013616599577097, 4.40850571664778979311498514023, 5.84512846361201473729238192484, 6.17059024064099961578062859224, 6.96043039935193165794878074137, 8.668579440672815407748615791918, 9.272741101030207950672752381230, 10.25786420242732647206052933090