| L(s) = 1 | − 1.47·2-s − 1.76·3-s + 0.174·4-s + 0.466·5-s + 2.59·6-s − 0.582·7-s + 2.69·8-s + 0.106·9-s − 0.688·10-s + 0.830·11-s − 0.307·12-s − 13-s + 0.858·14-s − 0.822·15-s − 4.31·16-s − 1.90·17-s − 0.157·18-s − 1.07·19-s + 0.0813·20-s + 1.02·21-s − 1.22·22-s + 1.37·23-s − 4.74·24-s − 4.78·25-s + 1.47·26-s + 5.09·27-s − 0.101·28-s + ⋯ |
| L(s) = 1 | − 1.04·2-s − 1.01·3-s + 0.0871·4-s + 0.208·5-s + 1.06·6-s − 0.220·7-s + 0.951·8-s + 0.0356·9-s − 0.217·10-s + 0.250·11-s − 0.0886·12-s − 0.277·13-s + 0.229·14-s − 0.212·15-s − 1.07·16-s − 0.462·17-s − 0.0371·18-s − 0.246·19-s + 0.0181·20-s + 0.223·21-s − 0.261·22-s + 0.286·23-s − 0.968·24-s − 0.956·25-s + 0.289·26-s + 0.981·27-s − 0.0191·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.47T + 2T^{2} \) |
| 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 - 0.466T + 5T^{2} \) |
| 7 | \( 1 + 0.582T + 7T^{2} \) |
| 11 | \( 1 - 0.830T + 11T^{2} \) |
| 17 | \( 1 + 1.90T + 17T^{2} \) |
| 19 | \( 1 + 1.07T + 19T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 - 9.83T + 29T^{2} \) |
| 31 | \( 1 - 4.32T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 - 5.02T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 - 8.25T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 8.81T + 61T^{2} \) |
| 67 | \( 1 - 2.39T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 - 2.22T + 79T^{2} \) |
| 83 | \( 1 - 0.783T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 2.29T + 97T^{2} \) |
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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
−9.294262100924081184476374713183, −8.493111949436629334621938053584, −7.74770652659057993276499754068, −6.67152213765947742428209645174, −6.12367075699860286207474314170, −4.98687071634487665910565479046, −4.30406477708512876679730705496, −2.68832613546223800123847662412, −1.22049367469249739787098907230, 0,
1.22049367469249739787098907230, 2.68832613546223800123847662412, 4.30406477708512876679730705496, 4.98687071634487665910565479046, 6.12367075699860286207474314170, 6.67152213765947742428209645174, 7.74770652659057993276499754068, 8.493111949436629334621938053584, 9.294262100924081184476374713183
