| L(s) = 1 | − 2.80·2-s + 3-s + 5.89·4-s + 3.04·5-s − 2.80·6-s + 3.35·7-s − 10.9·8-s + 9-s − 8.55·10-s + 3.20·11-s + 5.89·12-s + 6.42·13-s − 9.43·14-s + 3.04·15-s + 18.9·16-s − 6.70·17-s − 2.80·18-s + 5.19·19-s + 17.9·20-s + 3.35·21-s − 8.99·22-s − 1.38·23-s − 10.9·24-s + 4.26·25-s − 18.0·26-s + 27-s + 19.7·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 0.577·3-s + 2.94·4-s + 1.36·5-s − 1.14·6-s + 1.26·7-s − 3.87·8-s + 0.333·9-s − 2.70·10-s + 0.965·11-s + 1.70·12-s + 1.78·13-s − 2.52·14-s + 0.786·15-s + 4.74·16-s − 1.62·17-s − 0.662·18-s + 1.19·19-s + 4.01·20-s + 0.732·21-s − 1.91·22-s − 0.289·23-s − 2.23·24-s + 0.853·25-s − 3.54·26-s + 0.192·27-s + 3.74·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.086809567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.086809567\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 5 | \( 1 - 3.04T + 5T^{2} \) |
| 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 - 3.20T + 11T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 + 9.32T + 31T^{2} \) |
| 37 | \( 1 - 1.94T + 37T^{2} \) |
| 41 | \( 1 + 8.32T + 41T^{2} \) |
| 43 | \( 1 - 7.92T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 0.587T + 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 - 0.747T + 61T^{2} \) |
| 67 | \( 1 - 6.42T + 67T^{2} \) |
| 71 | \( 1 + 7.60T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 8.13T + 83T^{2} \) |
| 89 | \( 1 - 8.78T + 89T^{2} \) |
| 97 | \( 1 + 5.64T + 97T^{2} \) |
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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
−8.131420017024795365303534772031, −7.32127229188876930054851273583, −6.68598420633779376241000991017, −6.08377117910698094445141375580, −5.44411721363485455919708206093, −4.07498064223832005248277032097, −3.04675368907030910940542662783, −2.06027804214256810975194534760, −1.63258237879260752640122589798, −1.05246346788864707753994505661,
1.05246346788864707753994505661, 1.63258237879260752640122589798, 2.06027804214256810975194534760, 3.04675368907030910940542662783, 4.07498064223832005248277032097, 5.44411721363485455919708206093, 6.08377117910698094445141375580, 6.68598420633779376241000991017, 7.32127229188876930054851273583, 8.131420017024795365303534772031
