| L(s) = 1 | + 0.965·2-s − 1.06·4-s − 2.55·5-s − 0.0105·7-s − 2.96·8-s − 2.46·10-s + 1.39·11-s + 3.34·13-s − 0.0102·14-s − 0.723·16-s − 6.06·17-s + 19-s + 2.72·20-s + 1.34·22-s + 4.40·23-s + 1.52·25-s + 3.23·26-s + 0.0112·28-s − 1.61·29-s − 7.37·31-s + 5.22·32-s − 5.85·34-s + 0.0270·35-s − 5.34·37-s + 0.965·38-s + 7.56·40-s − 1.42·41-s + ⋯ |
| L(s) = 1 | + 0.682·2-s − 0.533·4-s − 1.14·5-s − 0.00399·7-s − 1.04·8-s − 0.780·10-s + 0.421·11-s + 0.927·13-s − 0.00272·14-s − 0.180·16-s − 1.47·17-s + 0.229·19-s + 0.610·20-s + 0.287·22-s + 0.919·23-s + 0.305·25-s + 0.633·26-s + 0.00213·28-s − 0.299·29-s − 1.32·31-s + 0.923·32-s − 1.00·34-s + 0.00456·35-s − 0.878·37-s + 0.156·38-s + 1.19·40-s − 0.222·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.154646228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.154646228\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.965T + 2T^{2} \) |
| 5 | \( 1 + 2.55T + 5T^{2} \) |
| 7 | \( 1 + 0.0105T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 - 3.34T + 13T^{2} \) |
| 17 | \( 1 + 6.06T + 17T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 + 1.61T + 29T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 + 5.34T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 3.31T + 59T^{2} \) |
| 61 | \( 1 - 2.27T + 61T^{2} \) |
| 67 | \( 1 + 2.20T + 67T^{2} \) |
| 71 | \( 1 - 8.50T + 71T^{2} \) |
| 73 | \( 1 + 2.83T + 73T^{2} \) |
| 79 | \( 1 - 4.59T + 79T^{2} \) |
| 83 | \( 1 + 0.102T + 83T^{2} \) |
| 89 | \( 1 - 1.59T + 89T^{2} \) |
| 97 | \( 1 - 7.59T + 97T^{2} \) |
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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
−7.895360220099879509185113737127, −6.92343759578559904171002565204, −6.56435232652093609247227085157, −5.52232969604722086886256579128, −4.95876260589371091699227618478, −4.13478269353836119700694101369, −3.71157770138589758233487524406, −3.10462511623136375524081237555, −1.80110075865705128786219187271, −0.47520757679991941490022512097,
0.47520757679991941490022512097, 1.80110075865705128786219187271, 3.10462511623136375524081237555, 3.71157770138589758233487524406, 4.13478269353836119700694101369, 4.95876260589371091699227618478, 5.52232969604722086886256579128, 6.56435232652093609247227085157, 6.92343759578559904171002565204, 7.895360220099879509185113737127
