| L(s) = 1 | + 2.25·2-s − 3.09·3-s + 3.08·4-s + 3.28·5-s − 6.98·6-s − 4.11·7-s + 2.43·8-s + 6.59·9-s + 7.39·10-s + 1.92·11-s − 9.54·12-s − 13-s − 9.26·14-s − 10.1·15-s − 0.664·16-s + 4.16·17-s + 14.8·18-s + 6.17·19-s + 10.1·20-s + 12.7·21-s + 4.33·22-s + 3.53·23-s − 7.55·24-s + 5.76·25-s − 2.25·26-s − 11.1·27-s − 12.6·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s − 1.78·3-s + 1.54·4-s + 1.46·5-s − 2.85·6-s − 1.55·7-s + 0.862·8-s + 2.19·9-s + 2.33·10-s + 0.579·11-s − 2.75·12-s − 0.277·13-s − 2.47·14-s − 2.62·15-s − 0.166·16-s + 1.00·17-s + 3.50·18-s + 1.41·19-s + 2.26·20-s + 2.77·21-s + 0.924·22-s + 0.737·23-s − 1.54·24-s + 1.15·25-s − 0.442·26-s − 2.14·27-s − 2.39·28-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
| \(L(1)\) |
\(\approx\) |
\(2.717646535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.717646535\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.25T + 2T^{2} \) |
| 3 | \( 1 + 3.09T + 3T^{2} \) |
| 5 | \( 1 - 3.28T + 5T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 - 1.92T + 11T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 - 6.17T + 19T^{2} \) |
| 23 | \( 1 - 3.53T + 23T^{2} \) |
| 29 | \( 1 - 5.33T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 7.75T + 37T^{2} \) |
| 41 | \( 1 - 2.76T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 9.59T + 47T^{2} \) |
| 53 | \( 1 + 6.81T + 53T^{2} \) |
| 59 | \( 1 - 8.39T + 59T^{2} \) |
| 61 | \( 1 + 2.27T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 0.175T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 + 9.50T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 - 6.65T + 97T^{2} \) |
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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
−9.812190176167661014553788846097, −9.392633602054466190626245412319, −7.18664910247768302145285012383, −6.55277757092743357170456159962, −6.08920611092159892634803575308, −5.47166818580510676227124555285, −4.94737332842543589473178512980, −3.71088180378690960225569879624, −2.72230202844668900162284151859, −1.10171140960915849007833698828,
1.10171140960915849007833698828, 2.72230202844668900162284151859, 3.71088180378690960225569879624, 4.94737332842543589473178512980, 5.47166818580510676227124555285, 6.08920611092159892634803575308, 6.55277757092743357170456159962, 7.18664910247768302145285012383, 9.392633602054466190626245412319, 9.812190176167661014553788846097
