| L(s) = 1 | − 2.28·2-s + 3.20·4-s − 2.49·5-s + 2.43·7-s − 2.76·8-s + 5.70·10-s + 5.17·11-s + 1.47·13-s − 5.55·14-s − 0.118·16-s − 4.99·17-s − 5.76·19-s − 8.01·20-s − 11.8·22-s + 6.91·23-s + 1.23·25-s − 3.36·26-s + 7.81·28-s + 29-s + 5.10·31-s + 5.79·32-s + 11.4·34-s − 6.08·35-s − 2.90·37-s + 13.1·38-s + 6.89·40-s − 10.1·41-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.60·4-s − 1.11·5-s + 0.920·7-s − 0.975·8-s + 1.80·10-s + 1.56·11-s + 0.408·13-s − 1.48·14-s − 0.0295·16-s − 1.21·17-s − 1.32·19-s − 1.79·20-s − 2.51·22-s + 1.44·23-s + 0.247·25-s − 0.659·26-s + 1.47·28-s + 0.185·29-s + 0.917·31-s + 1.02·32-s + 1.95·34-s − 1.02·35-s − 0.477·37-s + 2.13·38-s + 1.09·40-s − 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6444873547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6444873547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 5 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 + 4.99T + 17T^{2} \) |
| 19 | \( 1 + 5.76T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 + 2.90T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 9.06T + 43T^{2} \) |
| 47 | \( 1 - 4.65T + 47T^{2} \) |
| 53 | \( 1 - 1.45T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 9.32T + 61T^{2} \) |
| 67 | \( 1 - 2.35T + 67T^{2} \) |
| 71 | \( 1 - 2.26T + 71T^{2} \) |
| 73 | \( 1 - 6.93T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 - 6.71T + 83T^{2} \) |
| 89 | \( 1 - 9.39T + 89T^{2} \) |
| 97 | \( 1 + 4.57T + 97T^{2} \) |
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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
−10.30408212182499492033981522456, −9.062900716665539483182072402902, −8.667897539863079986297514624771, −8.063645868741080566130170899736, −7.00274868913131195823986679135, −6.53023328571961530259589857772, −4.69289822103640506110376032801, −3.85277565550369320477597857496, −2.10478902862660092739038175683, −0.866675451629629204438837002190,
0.866675451629629204438837002190, 2.10478902862660092739038175683, 3.85277565550369320477597857496, 4.69289822103640506110376032801, 6.53023328571961530259589857772, 7.00274868913131195823986679135, 8.063645868741080566130170899736, 8.667897539863079986297514624771, 9.062900716665539483182072402902, 10.30408212182499492033981522456
