| L(s) = 1 | − 2.19·2-s + 3.13·3-s + 2.82·4-s − 6.87·6-s − 7-s − 1.80·8-s + 6.81·9-s − 3.50·11-s + 8.83·12-s + 0.0428·13-s + 2.19·14-s − 1.68·16-s + 3.63·17-s − 14.9·18-s − 4.50·19-s − 3.13·21-s + 7.70·22-s − 23-s − 5.64·24-s − 0.0941·26-s + 11.9·27-s − 2.82·28-s − 8.72·29-s − 7.81·31-s + 7.30·32-s − 10.9·33-s − 7.97·34-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 1.80·3-s + 1.41·4-s − 2.80·6-s − 0.377·7-s − 0.636·8-s + 2.27·9-s − 1.05·11-s + 2.55·12-s + 0.0118·13-s + 0.586·14-s − 0.421·16-s + 0.880·17-s − 3.52·18-s − 1.03·19-s − 0.683·21-s + 1.64·22-s − 0.208·23-s − 1.15·24-s − 0.0184·26-s + 2.30·27-s − 0.532·28-s − 1.61·29-s − 1.40·31-s + 1.29·32-s − 1.91·33-s − 1.36·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 3 | \( 1 - 3.13T + 3T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 - 0.0428T + 13T^{2} \) |
| 17 | \( 1 - 3.63T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 29 | \( 1 + 8.72T + 29T^{2} \) |
| 31 | \( 1 + 7.81T + 31T^{2} \) |
| 37 | \( 1 + 1.38T + 37T^{2} \) |
| 41 | \( 1 + 0.960T + 41T^{2} \) |
| 43 | \( 1 + 2.92T + 43T^{2} \) |
| 47 | \( 1 + 3.94T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 + 0.660T + 61T^{2} \) |
| 67 | \( 1 - 8.15T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.164678336112761145957293215281, −7.66247465844853504695114697395, −7.24444560564765833709738948384, −6.24933921497433409180956581358, −5.00389217312685682495653649994, −3.83928239676549482998250847810, −3.12494981721676917088353123103, −2.20046146382143413283255986867, −1.62257653430730342317708999750, 0,
1.62257653430730342317708999750, 2.20046146382143413283255986867, 3.12494981721676917088353123103, 3.83928239676549482998250847810, 5.00389217312685682495653649994, 6.24933921497433409180956581358, 7.24444560564765833709738948384, 7.66247465844853504695114697395, 8.164678336112761145957293215281
