| L(s) = 1 | − 2.58·2-s − 2.62·3-s + 4.69·4-s + 5-s + 6.78·6-s − 4.29·7-s − 6.96·8-s + 3.87·9-s − 2.58·10-s + 0.392·11-s − 12.3·12-s + 3.86·13-s + 11.1·14-s − 2.62·15-s + 8.63·16-s + 2.28·17-s − 10.0·18-s − 6.51·19-s + 4.69·20-s + 11.2·21-s − 1.01·22-s − 3.95·23-s + 18.2·24-s + 25-s − 9.99·26-s − 2.30·27-s − 20.1·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 1.51·3-s + 2.34·4-s + 0.447·5-s + 2.76·6-s − 1.62·7-s − 2.46·8-s + 1.29·9-s − 0.818·10-s + 0.118·11-s − 3.55·12-s + 1.07·13-s + 2.97·14-s − 0.677·15-s + 2.15·16-s + 0.553·17-s − 2.36·18-s − 1.49·19-s + 1.04·20-s + 2.45·21-s − 0.216·22-s − 0.825·23-s + 3.72·24-s + 0.200·25-s − 1.95·26-s − 0.442·27-s − 3.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 + 2.62T + 3T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 11 | \( 1 - 0.392T + 11T^{2} \) |
| 13 | \( 1 - 3.86T + 13T^{2} \) |
| 17 | \( 1 - 2.28T + 17T^{2} \) |
| 19 | \( 1 + 6.51T + 19T^{2} \) |
| 23 | \( 1 + 3.95T + 23T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 - 0.970T + 31T^{2} \) |
| 37 | \( 1 - 3.07T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 0.970T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 + 8.72T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 7.87T + 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 - 1.20T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 6.20T + 97T^{2} \) |
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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
−9.425262205915982946528702008620, −8.822942537881689446303440870778, −7.68554830854600174603810495047, −6.72783399121897076265505420238, −6.12589170908951824683443406967, −5.87104038972693220338769474009, −3.99088308163490814385611339770, −2.52508793443217437571207566743, −1.08664086166235773481388863198, 0,
1.08664086166235773481388863198, 2.52508793443217437571207566743, 3.99088308163490814385611339770, 5.87104038972693220338769474009, 6.12589170908951824683443406967, 6.72783399121897076265505420238, 7.68554830854600174603810495047, 8.822942537881689446303440870778, 9.425262205915982946528702008620
