| L(s) = 1 | + 2-s − 3-s + 4-s + 0.926·5-s − 6-s − 0.782·7-s + 8-s + 9-s + 0.926·10-s − 4.19·11-s − 12-s + 1.67·13-s − 0.782·14-s − 0.926·15-s + 16-s + 0.331·17-s + 18-s − 19-s + 0.926·20-s + 0.782·21-s − 4.19·22-s + 0.149·23-s − 24-s − 4.14·25-s + 1.67·26-s − 27-s − 0.782·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.414·5-s − 0.408·6-s − 0.295·7-s + 0.353·8-s + 0.333·9-s + 0.293·10-s − 1.26·11-s − 0.288·12-s + 0.464·13-s − 0.209·14-s − 0.239·15-s + 0.250·16-s + 0.0804·17-s + 0.235·18-s − 0.229·19-s + 0.207·20-s + 0.170·21-s − 0.895·22-s + 0.0312·23-s − 0.204·24-s − 0.828·25-s + 0.328·26-s − 0.192·27-s − 0.147·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 - 0.926T + 5T^{2} \) |
| 7 | \( 1 + 0.782T + 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 - 1.67T + 13T^{2} \) |
| 17 | \( 1 - 0.331T + 17T^{2} \) |
| 23 | \( 1 - 0.149T + 23T^{2} \) |
| 29 | \( 1 - 3.09T + 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 5.83T + 41T^{2} \) |
| 43 | \( 1 - 6.59T + 43T^{2} \) |
| 47 | \( 1 + 0.211T + 47T^{2} \) |
| 59 | \( 1 + 9.04T + 59T^{2} \) |
| 61 | \( 1 + 0.848T + 61T^{2} \) |
| 67 | \( 1 - 1.44T + 67T^{2} \) |
| 71 | \( 1 + 8.46T + 71T^{2} \) |
| 73 | \( 1 + 8.41T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.66506449893756919111422796137, −6.70490626392944288737559838702, −6.26153228168184799045870506220, −5.45970305299765832399870514112, −5.03171034260570210838448524669, −4.14285335476980460139079180284, −3.25311133706435193273410803397, −2.44227573511795468576887751662, −1.45768648609975416125077409102, 0,
1.45768648609975416125077409102, 2.44227573511795468576887751662, 3.25311133706435193273410803397, 4.14285335476980460139079180284, 5.03171034260570210838448524669, 5.45970305299765832399870514112, 6.26153228168184799045870506220, 6.70490626392944288737559838702, 7.66506449893756919111422796137
