| L(s) = 1 | + 0.818·2-s − 3·3-s − 7.32·4-s − 5·5-s − 2.45·6-s + 13.8·7-s − 12.5·8-s + 9·9-s − 4.09·10-s + 71.7·11-s + 21.9·12-s − 78.8·13-s + 11.3·14-s + 15·15-s + 48.3·16-s + 93.8·17-s + 7.36·18-s − 15.4·19-s + 36.6·20-s − 41.5·21-s + 58.7·22-s − 187.·23-s + 37.6·24-s + 25·25-s − 64.5·26-s − 27·27-s − 101.·28-s + ⋯ |
| L(s) = 1 | + 0.289·2-s − 0.577·3-s − 0.916·4-s − 0.447·5-s − 0.167·6-s + 0.748·7-s − 0.554·8-s + 0.333·9-s − 0.129·10-s + 1.96·11-s + 0.528·12-s − 1.68·13-s + 0.216·14-s + 0.258·15-s + 0.755·16-s + 1.33·17-s + 0.0964·18-s − 0.186·19-s + 0.409·20-s − 0.432·21-s + 0.569·22-s − 1.69·23-s + 0.320·24-s + 0.200·25-s − 0.487·26-s − 0.192·27-s − 0.685·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 0.818T + 8T^{2} \) |
| 7 | \( 1 - 13.8T + 343T^{2} \) |
| 11 | \( 1 - 71.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 187.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 73.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 127.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 124.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 41.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 588.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 125.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 55.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 259.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 664.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 265.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 5.27T + 3.89e5T^{2} \) |
| 79 | \( 1 + 792.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.42e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 764.T + 9.12e5T^{2} \) |
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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
−10.08274129967376189151901062749, −9.549088617473806412226396891444, −8.405187943380156907984833996804, −7.53626120969845468028908745958, −6.36765691704215368758715813389, −5.24237162854773291860623375510, −4.45355835617734043988422839818, −3.57727661138514481001836790764, −1.49299544987160675292208844514, 0,
1.49299544987160675292208844514, 3.57727661138514481001836790764, 4.45355835617734043988422839818, 5.24237162854773291860623375510, 6.36765691704215368758715813389, 7.53626120969845468028908745958, 8.405187943380156907984833996804, 9.549088617473806412226396891444, 10.08274129967376189151901062749
