L(s) = 1 | − 2-s + 4-s − 0.137·5-s + 3.14·7-s − 8-s + 0.137·10-s − 1.47·11-s − 1.75·13-s − 3.14·14-s + 16-s + 4.07·17-s + 1.58·19-s − 0.137·20-s + 1.47·22-s − 8.54·23-s − 4.98·25-s + 1.75·26-s + 3.14·28-s − 1.49·29-s + 0.711·31-s − 32-s − 4.07·34-s − 0.431·35-s − 6.13·37-s − 1.58·38-s + 0.137·40-s − 4.46·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.0613·5-s + 1.18·7-s − 0.353·8-s + 0.0433·10-s − 0.445·11-s − 0.487·13-s − 0.839·14-s + 0.250·16-s + 0.989·17-s + 0.364·19-s − 0.0306·20-s + 0.314·22-s − 1.78·23-s − 0.996·25-s + 0.344·26-s + 0.593·28-s − 0.278·29-s + 0.127·31-s − 0.176·32-s − 0.699·34-s − 0.0728·35-s − 1.00·37-s − 0.257·38-s + 0.0216·40-s − 0.696·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
good | 5 | \( 1 + 0.137T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 + 1.75T + 13T^{2} \) |
| 17 | \( 1 - 4.07T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 + 8.54T + 23T^{2} \) |
| 29 | \( 1 + 1.49T + 29T^{2} \) |
| 31 | \( 1 - 0.711T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 + 4.46T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 + 3.50T + 47T^{2} \) |
| 53 | \( 1 - 8.57T + 53T^{2} \) |
| 59 | \( 1 - 0.0537T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 6.84T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 - 7.44T + 73T^{2} \) |
| 79 | \( 1 + 5.85T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 2.93T + 97T^{2} \) |
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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
−8.003665894550952590561423350326, −7.68316958799596074609354067383, −6.83082082958628927289880567163, −5.75459562350026494121288162114, −5.25836335739556313167362824485, −4.27182428492876514093573159049, −3.31375444670676551741619182032, −2.15910794996260776734890872406, −1.47888940640797339773904973538, 0,
1.47888940640797339773904973538, 2.15910794996260776734890872406, 3.31375444670676551741619182032, 4.27182428492876514093573159049, 5.25836335739556313167362824485, 5.75459562350026494121288162114, 6.83082082958628927289880567163, 7.68316958799596074609354067383, 8.003665894550952590561423350326