| L(s) = 1 | − 182.·2-s − 3.00e3·3-s + 589.·4-s + 3.73e4·5-s + 5.48e5·6-s + 4.15e6·7-s + 5.87e6·8-s − 5.33e6·9-s − 6.81e6·10-s + 2.55e7·11-s − 1.76e6·12-s + 3.68e8·13-s − 7.59e8·14-s − 1.12e8·15-s − 1.09e9·16-s − 8.33e8·17-s + 9.74e8·18-s + 6.37e9·19-s + 2.19e7·20-s − 1.24e10·21-s − 4.67e9·22-s − 2.63e10·23-s − 1.76e10·24-s − 2.91e10·25-s − 6.72e10·26-s + 5.90e10·27-s + 2.45e9·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.792·3-s + 0.0179·4-s + 0.213·5-s + 0.799·6-s + 1.90·7-s + 0.990·8-s − 0.371·9-s − 0.215·10-s + 0.395·11-s − 0.0142·12-s + 1.62·13-s − 1.92·14-s − 0.169·15-s − 1.01·16-s − 0.492·17-s + 0.375·18-s + 1.63·19-s + 0.00384·20-s − 1.51·21-s − 0.399·22-s − 1.61·23-s − 0.785·24-s − 0.954·25-s − 1.64·26-s + 1.08·27-s + 0.0343·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 - 1.22e14T \) |
| good | 2 | \( 1 + 182.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 3.00e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 3.73e4T + 3.05e10T^{2} \) |
| 7 | \( 1 - 4.15e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 2.55e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.68e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 8.33e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 6.37e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 2.63e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 1.70e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 2.38e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 3.20e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 1.37e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.78e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 1.06e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 1.22e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 4.26e12T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.24e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 3.59e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 9.91e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.67e14T + 8.90e27T^{2} \) |
| 79 | \( 1 - 4.17e13T + 2.91e28T^{2} \) |
| 83 | \( 1 + 1.73e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 9.08e13T + 1.74e29T^{2} \) |
| 97 | \( 1 - 3.05e14T + 6.33e29T^{2} \) |
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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
−10.59882505779391222115709611093, −9.210198420015831465698560313682, −8.350230463005268409881942077061, −7.57734602241187655469040530059, −5.97373219653643366828005255089, −5.08788622180618264943415462203, −3.93220949169342907050038738824, −1.74879841992653285243019458875, −1.22462444998863624686918983490, 0,
1.22462444998863624686918983490, 1.74879841992653285243019458875, 3.93220949169342907050038738824, 5.08788622180618264943415462203, 5.97373219653643366828005255089, 7.57734602241187655469040530059, 8.350230463005268409881942077061, 9.210198420015831465698560313682, 10.59882505779391222115709611093
