| L(s) = 1 | + 2-s + 0.618·3-s + 4-s + 0.618·5-s + 0.618·6-s − 2.85·7-s + 8-s − 2.61·9-s + 0.618·10-s + 0.618·12-s + 1.61·13-s − 2.85·14-s + 0.381·15-s + 16-s + 1.23·17-s − 2.61·18-s + 19-s + 0.618·20-s − 1.76·21-s − 2·23-s + 0.618·24-s − 4.61·25-s + 1.61·26-s − 3.47·27-s − 2.85·28-s − 5.85·29-s + 0.381·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.356·3-s + 0.5·4-s + 0.276·5-s + 0.252·6-s − 1.07·7-s + 0.353·8-s − 0.872·9-s + 0.195·10-s + 0.178·12-s + 0.448·13-s − 0.762·14-s + 0.0986·15-s + 0.250·16-s + 0.299·17-s − 0.617·18-s + 0.229·19-s + 0.138·20-s − 0.384·21-s − 0.417·23-s + 0.126·24-s − 0.923·25-s + 0.317·26-s − 0.668·27-s − 0.539·28-s − 1.08·29-s + 0.0697·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 + 0.381T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 1.61T + 41T^{2} \) |
| 43 | \( 1 + 0.909T + 43T^{2} \) |
| 47 | \( 1 + 5.23T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 - 6.14T + 71T^{2} \) |
| 73 | \( 1 + 1.52T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 4.61T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 9.52T + 97T^{2} \) |
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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
−7.929007849642636159687292584250, −7.12102679703490535698028213871, −6.28485070753969783161114021682, −5.82217423965232809617317473378, −5.13804168187903940118684905143, −3.91313313259222151379392665912, −3.41550729580287004738711400103, −2.66430133731107815385021548385, −1.69030125648353964306231340091, 0,
1.69030125648353964306231340091, 2.66430133731107815385021548385, 3.41550729580287004738711400103, 3.91313313259222151379392665912, 5.13804168187903940118684905143, 5.82217423965232809617317473378, 6.28485070753969783161114021682, 7.12102679703490535698028213871, 7.929007849642636159687292584250
