| L(s) = 1 | + 6.55e4·2-s − 3.44e6·3-s + 4.29e9·4-s + 1.52e11·5-s − 2.25e11·6-s + 9.65e13·7-s + 2.81e14·8-s − 5.54e15·9-s + 1.00e16·10-s − 1.25e17·11-s − 1.47e16·12-s − 3.83e18·13-s + 6.33e18·14-s − 5.25e17·15-s + 1.84e19·16-s − 1.12e19·17-s − 3.63e20·18-s − 1.21e21·19-s + 6.55e20·20-s − 3.32e20·21-s − 8.23e21·22-s + 3.30e22·23-s − 9.68e20·24-s + 2.32e22·25-s − 2.51e23·26-s + 3.82e22·27-s + 4.14e23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0461·3-s + 0.5·4-s + 0.447·5-s − 0.0326·6-s + 1.09·7-s + 0.353·8-s − 0.997·9-s + 0.316·10-s − 0.824·11-s − 0.0230·12-s − 1.59·13-s + 0.776·14-s − 0.0206·15-s + 0.250·16-s − 0.0561·17-s − 0.705·18-s − 0.967·19-s + 0.223·20-s − 0.0506·21-s − 0.582·22-s + 1.12·23-s − 0.0163·24-s + 0.200·25-s − 1.13·26-s + 0.0921·27-s + 0.549·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 6.55e4T \) |
| 5 | \( 1 - 1.52e11T \) |
| good | 3 | \( 1 + 3.44e6T + 5.55e15T^{2} \) |
| 7 | \( 1 - 9.65e13T + 7.73e27T^{2} \) |
| 11 | \( 1 + 1.25e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 3.83e18T + 5.75e36T^{2} \) |
| 17 | \( 1 + 1.12e19T + 4.02e40T^{2} \) |
| 19 | \( 1 + 1.21e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 3.30e22T + 8.65e44T^{2} \) |
| 29 | \( 1 - 5.37e23T + 1.81e48T^{2} \) |
| 31 | \( 1 + 4.22e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 1.37e26T + 5.63e51T^{2} \) |
| 41 | \( 1 - 4.39e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 1.20e27T + 8.02e53T^{2} \) |
| 47 | \( 1 + 3.56e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 3.20e28T + 7.96e56T^{2} \) |
| 59 | \( 1 - 2.08e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 1.84e29T + 8.23e58T^{2} \) |
| 67 | \( 1 + 3.61e29T + 1.82e60T^{2} \) |
| 71 | \( 1 + 5.64e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 6.52e29T + 3.08e61T^{2} \) |
| 79 | \( 1 + 2.61e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 7.88e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 6.51e31T + 2.13e64T^{2} \) |
| 97 | \( 1 + 7.62e32T + 3.65e65T^{2} \) |
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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
−12.89383625080987991795024700497, −11.58192177592124405692874893732, −10.42347198496466689586768067043, −8.573097156331468292096075667245, −7.17559848908568966657586865605, −5.48695693138631109521790328039, −4.76759483324653587227046770781, −2.85631086388946562924965818818, −1.88134754992078329305752093878, 0,
1.88134754992078329305752093878, 2.85631086388946562924965818818, 4.76759483324653587227046770781, 5.48695693138631109521790328039, 7.17559848908568966657586865605, 8.573097156331468292096075667245, 10.42347198496466689586768067043, 11.58192177592124405692874893732, 12.89383625080987991795024700497
