| L(s) = 1 | − 2-s + 0.809·3-s + 4-s + 3.41·5-s − 0.809·6-s − 7-s − 8-s − 2.34·9-s − 3.41·10-s + 2.19·11-s + 0.809·12-s + 1.00·13-s + 14-s + 2.76·15-s + 16-s − 5.35·17-s + 2.34·18-s − 3.13·19-s + 3.41·20-s − 0.809·21-s − 2.19·22-s − 6.05·23-s − 0.809·24-s + 6.64·25-s − 1.00·26-s − 4.32·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.467·3-s + 0.5·4-s + 1.52·5-s − 0.330·6-s − 0.377·7-s − 0.353·8-s − 0.781·9-s − 1.07·10-s + 0.663·11-s + 0.233·12-s + 0.277·13-s + 0.267·14-s + 0.713·15-s + 0.250·16-s − 1.29·17-s + 0.552·18-s − 0.718·19-s + 0.763·20-s − 0.176·21-s − 0.468·22-s − 1.26·23-s − 0.165·24-s + 1.32·25-s − 0.196·26-s − 0.832·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 0.809T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 + 5.35T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 23 | \( 1 + 6.05T + 23T^{2} \) |
| 29 | \( 1 - 0.210T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 - 7.78T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 2.20T + 71T^{2} \) |
| 73 | \( 1 + 4.98T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 0.148T + 89T^{2} \) |
| 97 | \( 1 - 4.76T + 97T^{2} \) |
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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
−8.003624836464453271579988813101, −6.85087658978419619349759278847, −6.25486887360485679428208933430, −6.01031261085636703709109245849, −4.93367449865681277355177127000, −3.89085039271895496079882886014, −2.88283107818091678893375296206, −2.18882492997397179741155454001, −1.55343390598840517263287609031, 0,
1.55343390598840517263287609031, 2.18882492997397179741155454001, 2.88283107818091678893375296206, 3.89085039271895496079882886014, 4.93367449865681277355177127000, 6.01031261085636703709109245849, 6.25486887360485679428208933430, 6.85087658978419619349759278847, 8.003624836464453271579988813101
