| L(s) = 1 | + 0.919·2-s + 1.18·3-s − 1.15·4-s − 3.27·5-s + 1.09·6-s − 0.836·7-s − 2.90·8-s − 1.59·9-s − 3.00·10-s − 5.47·11-s − 1.36·12-s − 4.17·13-s − 0.769·14-s − 3.88·15-s − 0.359·16-s − 2.47·17-s − 1.46·18-s − 19-s + 3.77·20-s − 0.991·21-s − 5.03·22-s − 3.40·23-s − 3.43·24-s + 5.71·25-s − 3.83·26-s − 5.44·27-s + 0.965·28-s + ⋯ |
| L(s) = 1 | + 0.650·2-s + 0.684·3-s − 0.577·4-s − 1.46·5-s + 0.445·6-s − 0.316·7-s − 1.02·8-s − 0.531·9-s − 0.951·10-s − 1.65·11-s − 0.395·12-s − 1.15·13-s − 0.205·14-s − 1.00·15-s − 0.0898·16-s − 0.600·17-s − 0.345·18-s − 0.229·19-s + 0.844·20-s − 0.216·21-s − 1.07·22-s − 0.709·23-s − 0.702·24-s + 1.14·25-s − 0.752·26-s − 1.04·27-s + 0.182·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05965128995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05965128995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 - 0.919T + 2T^{2} \) |
| 3 | \( 1 - 1.18T + 3T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 + 0.836T + 7T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 23 | \( 1 + 3.40T + 23T^{2} \) |
| 29 | \( 1 + 7.56T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 6.66T + 37T^{2} \) |
| 41 | \( 1 + 5.63T + 41T^{2} \) |
| 43 | \( 1 + 9.97T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 0.122T + 59T^{2} \) |
| 61 | \( 1 + 1.95T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 - 1.72T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 + 0.284T + 79T^{2} \) |
| 83 | \( 1 - 5.33T + 83T^{2} \) |
| 89 | \( 1 + 0.788T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.984404313049829978139804616385, −7.73924488031848664778465774470, −6.68789315806927967936788636796, −5.74939500116518248989863632648, −4.93871311451754579559000329546, −4.45449201015688027360109259345, −3.61606760355112589763357862268, −2.96962461238087995102451223832, −2.37639364664868867222701114753, −0.10243873959005800430867364947,
0.10243873959005800430867364947, 2.37639364664868867222701114753, 2.96962461238087995102451223832, 3.61606760355112589763357862268, 4.45449201015688027360109259345, 4.93871311451754579559000329546, 5.74939500116518248989863632648, 6.68789315806927967936788636796, 7.73924488031848664778465774470, 7.984404313049829978139804616385
