| L(s) = 1 | − 0.345·2-s − 1.88·4-s − 3.42·5-s + 1.33·8-s + 1.18·10-s + 1.68·11-s − 13-s + 3.29·16-s + 4.05·17-s − 6.06·19-s + 6.43·20-s − 0.581·22-s − 5.79·23-s + 6.70·25-s + 0.345·26-s + 6.48·29-s − 0.299·31-s − 3.81·32-s − 1.40·34-s + 7.76·37-s + 2.09·38-s − 4.58·40-s − 9.90·41-s − 6.06·43-s − 3.16·44-s + 2·46-s − 7.47·47-s + ⋯ |
| L(s) = 1 | − 0.244·2-s − 0.940·4-s − 1.52·5-s + 0.473·8-s + 0.373·10-s + 0.507·11-s − 0.277·13-s + 0.824·16-s + 0.984·17-s − 1.39·19-s + 1.43·20-s − 0.123·22-s − 1.20·23-s + 1.34·25-s + 0.0676·26-s + 1.20·29-s − 0.0538·31-s − 0.674·32-s − 0.240·34-s + 1.27·37-s + 0.339·38-s − 0.724·40-s − 1.54·41-s − 0.924·43-s − 0.477·44-s + 0.294·46-s − 1.09·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5093034970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5093034970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.345T + 2T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 17 | \( 1 - 4.05T + 17T^{2} \) |
| 19 | \( 1 + 6.06T + 19T^{2} \) |
| 23 | \( 1 + 5.79T + 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 + 0.299T + 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 + 6.06T + 43T^{2} \) |
| 47 | \( 1 + 7.47T + 47T^{2} \) |
| 53 | \( 1 + 9.85T + 53T^{2} \) |
| 59 | \( 1 + 3.75T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 9.76T + 67T^{2} \) |
| 71 | \( 1 + 8.52T + 71T^{2} \) |
| 73 | \( 1 - 1.70T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 0.638T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 4.06T + 97T^{2} \) |
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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
−8.171945777595921297040906314128, −7.75353892506109922947009352262, −6.78201192996195302320712422428, −6.06840615511541843087362464704, −4.90452196321672816527513546995, −4.44876886363588919470704067776, −3.76798727405518498682421044930, −3.11049317919326886550711285939, −1.60952258780521424618542289425, −0.40611662348571470060544194006,
0.40611662348571470060544194006, 1.60952258780521424618542289425, 3.11049317919326886550711285939, 3.76798727405518498682421044930, 4.44876886363588919470704067776, 4.90452196321672816527513546995, 6.06840615511541843087362464704, 6.78201192996195302320712422428, 7.75353892506109922947009352262, 8.171945777595921297040906314128
