| L(s) = 1 | − 2-s − 1.10·3-s + 4-s − 0.0681·5-s + 1.10·6-s + 2.43·7-s − 8-s − 1.78·9-s + 0.0681·10-s + 5.61·11-s − 1.10·12-s + 1.70·13-s − 2.43·14-s + 0.0751·15-s + 16-s − 3.58·17-s + 1.78·18-s − 4.83·19-s − 0.0681·20-s − 2.68·21-s − 5.61·22-s + 4.79·23-s + 1.10·24-s − 4.99·25-s − 1.70·26-s + 5.27·27-s + 2.43·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.637·3-s + 0.5·4-s − 0.0304·5-s + 0.450·6-s + 0.920·7-s − 0.353·8-s − 0.594·9-s + 0.0215·10-s + 1.69·11-s − 0.318·12-s + 0.471·13-s − 0.651·14-s + 0.0194·15-s + 0.250·16-s − 0.870·17-s + 0.420·18-s − 1.10·19-s − 0.0152·20-s − 0.586·21-s − 1.19·22-s + 0.999·23-s + 0.225·24-s − 0.999·25-s − 0.333·26-s + 1.01·27-s + 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.001450891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001450891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 1.10T + 3T^{2} \) |
| 5 | \( 1 + 0.0681T + 5T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 - 5.61T + 11T^{2} \) |
| 13 | \( 1 - 1.70T + 13T^{2} \) |
| 17 | \( 1 + 3.58T + 17T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 23 | \( 1 - 4.79T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 - 3.27T + 37T^{2} \) |
| 41 | \( 1 - 4.85T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 8.98T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 1.99T + 71T^{2} \) |
| 73 | \( 1 - 6.42T + 73T^{2} \) |
| 79 | \( 1 - 6.23T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 9.89T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−10.33525570528387829556833574922, −8.936959347578442996907878887889, −8.827231399344623107919346487931, −7.71659126151810509269699017379, −6.56155370613151692581545383749, −6.13926162462006156762536054053, −4.86506488943449905879590807788, −3.87525105139685593427920287695, −2.24791839778951874881219456553, −0.963170104323067709732555299310,
0.963170104323067709732555299310, 2.24791839778951874881219456553, 3.87525105139685593427920287695, 4.86506488943449905879590807788, 6.13926162462006156762536054053, 6.56155370613151692581545383749, 7.71659126151810509269699017379, 8.827231399344623107919346487931, 8.936959347578442996907878887889, 10.33525570528387829556833574922
