| L(s) = 1 | − 2.52·2-s + 4.39·4-s − 0.133·5-s − 7-s − 6.05·8-s + 0.337·10-s − 11-s + 0.133·13-s + 2.52·14-s + 6.52·16-s + 5.05·17-s − 0.924·19-s − 0.586·20-s + 2.52·22-s + 7.05·23-s − 4.98·25-s − 0.337·26-s − 4.39·28-s − 3.86·29-s + 2.79·31-s − 4.39·32-s − 12.7·34-s + 0.133·35-s + 9.98·37-s + 2.33·38-s + 0.808·40-s − 11.8·41-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 2.19·4-s − 0.0596·5-s − 0.377·7-s − 2.14·8-s + 0.106·10-s − 0.301·11-s + 0.0370·13-s + 0.675·14-s + 1.63·16-s + 1.22·17-s − 0.212·19-s − 0.131·20-s + 0.539·22-s + 1.47·23-s − 0.996·25-s − 0.0662·26-s − 0.830·28-s − 0.717·29-s + 0.501·31-s − 0.777·32-s − 2.19·34-s + 0.0225·35-s + 1.64·37-s + 0.379·38-s + 0.127·40-s − 1.85·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5931917339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5931917339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 5 | \( 1 + 0.133T + 5T^{2} \) |
| 13 | \( 1 - 0.133T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + 0.924T + 19T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 + 3.86T + 29T^{2} \) |
| 31 | \( 1 - 2.79T + 31T^{2} \) |
| 37 | \( 1 - 9.98T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 3.05T + 43T^{2} \) |
| 47 | \( 1 - 3.07T + 47T^{2} \) |
| 53 | \( 1 - 4.79T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 8.92T + 67T^{2} \) |
| 71 | \( 1 - 6.11T + 71T^{2} \) |
| 73 | \( 1 - 7.86T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 1.20T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−10.06287083629663070256959805184, −9.742622373161863821386341305810, −8.751931211758680633962335380238, −8.016988979946331964722263858808, −7.26462857320688202185871362338, −6.42103710521066609766729256108, −5.30200701719432773777941487258, −3.50835113085756827866004396710, −2.29477518003028835573674020964, −0.849121879007399067292849754039,
0.849121879007399067292849754039, 2.29477518003028835573674020964, 3.50835113085756827866004396710, 5.30200701719432773777941487258, 6.42103710521066609766729256108, 7.26462857320688202185871362338, 8.016988979946331964722263858808, 8.751931211758680633962335380238, 9.742622373161863821386341305810, 10.06287083629663070256959805184
