L(s) = 1 | + 2-s + 3-s + 4-s + 1.14·5-s + 6-s − 1.54·7-s + 8-s + 9-s + 1.14·10-s − 11-s + 12-s − 6.01·13-s − 1.54·14-s + 1.14·15-s + 16-s − 5.63·17-s + 18-s − 2.94·19-s + 1.14·20-s − 1.54·21-s − 22-s − 3.47·23-s + 24-s − 3.69·25-s − 6.01·26-s + 27-s − 1.54·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.511·5-s + 0.408·6-s − 0.584·7-s + 0.353·8-s + 0.333·9-s + 0.361·10-s − 0.301·11-s + 0.288·12-s − 1.66·13-s − 0.413·14-s + 0.295·15-s + 0.250·16-s − 1.36·17-s + 0.235·18-s − 0.675·19-s + 0.255·20-s − 0.337·21-s − 0.213·22-s − 0.725·23-s + 0.204·24-s − 0.738·25-s − 1.18·26-s + 0.192·27-s − 0.292·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 5 | \( 1 - 1.14T + 5T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 13 | \( 1 + 6.01T + 13T^{2} \) |
| 17 | \( 1 + 5.63T + 17T^{2} \) |
| 19 | \( 1 + 2.94T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 + 2.10T + 29T^{2} \) |
| 31 | \( 1 + 4.77T + 31T^{2} \) |
| 37 | \( 1 - 6.21T + 37T^{2} \) |
| 41 | \( 1 + 1.67T + 41T^{2} \) |
| 43 | \( 1 - 5.01T + 43T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 + 6.35T + 59T^{2} \) |
| 67 | \( 1 - 1.66T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 - 3.82T + 73T^{2} \) |
| 79 | \( 1 - 2.20T + 79T^{2} \) |
| 83 | \( 1 + 4.41T + 83T^{2} \) |
| 89 | \( 1 - 1.84T + 89T^{2} \) |
| 97 | \( 1 - 8.66T + 97T^{2} \) |
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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
−7.897901321236244994841813858891, −7.31167440103198892209576944858, −6.51843832369090079520249354499, −5.90557040180825552084621218104, −4.92976604872709052147544154572, −4.31013703724824540958673281826, −3.39921135594231060865697817838, −2.34506872001969458374719879997, −2.07374107656679427706389619976, 0,
2.07374107656679427706389619976, 2.34506872001969458374719879997, 3.39921135594231060865697817838, 4.31013703724824540958673281826, 4.92976604872709052147544154572, 5.90557040180825552084621218104, 6.51843832369090079520249354499, 7.31167440103198892209576944858, 7.897901321236244994841813858891