| L(s) = 1 | − 2-s − 1.17·3-s + 4-s − 3.87·5-s + 1.17·6-s − 1.06·7-s − 8-s − 1.61·9-s + 3.87·10-s + 4.42·11-s − 1.17·12-s + 1.85·13-s + 1.06·14-s + 4.55·15-s + 16-s + 5.59·17-s + 1.61·18-s + 0.300·19-s − 3.87·20-s + 1.24·21-s − 4.42·22-s + 23-s + 1.17·24-s + 10.0·25-s − 1.85·26-s + 5.43·27-s − 1.06·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.679·3-s + 0.5·4-s − 1.73·5-s + 0.480·6-s − 0.401·7-s − 0.353·8-s − 0.538·9-s + 1.22·10-s + 1.33·11-s − 0.339·12-s + 0.514·13-s + 0.283·14-s + 1.17·15-s + 0.250·16-s + 1.35·17-s + 0.380·18-s + 0.0688·19-s − 0.866·20-s + 0.272·21-s − 0.943·22-s + 0.208·23-s + 0.240·24-s + 2.00·25-s − 0.363·26-s + 1.04·27-s − 0.200·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 1.17T + 3T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 + 1.06T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 - 5.59T + 17T^{2} \) |
| 19 | \( 1 - 0.300T + 19T^{2} \) |
| 31 | \( 1 + 6.83T + 31T^{2} \) |
| 37 | \( 1 + 8.42T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 1.82T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 0.523T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 2.77T + 67T^{2} \) |
| 71 | \( 1 + 9.91T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 3.57T + 79T^{2} \) |
| 83 | \( 1 - 6.28T + 83T^{2} \) |
| 89 | \( 1 + 0.624T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−9.000313958886327668317153913687, −8.495918064853217092874817029458, −7.59900033049437771527413374662, −6.89560087121520087825184772171, −6.10104838859536239741567484487, −5.03972504405646154719767628873, −3.73969192555920505482568885472, −3.27369955456161043831708398565, −1.21077866954330363826115610317, 0,
1.21077866954330363826115610317, 3.27369955456161043831708398565, 3.73969192555920505482568885472, 5.03972504405646154719767628873, 6.10104838859536239741567484487, 6.89560087121520087825184772171, 7.59900033049437771527413374662, 8.495918064853217092874817029458, 9.000313958886327668317153913687
