| L(s) = 1 | + 1.89·2-s + 1.60·4-s + 5-s + 2.75·7-s − 0.744·8-s + 1.89·10-s + 3.41·11-s + 3.95·13-s + 5.22·14-s − 4.63·16-s − 4.75·17-s + 8.61·19-s + 1.60·20-s + 6.48·22-s + 0.0307·23-s + 25-s + 7.51·26-s + 4.42·28-s − 6.50·29-s − 4.13·31-s − 7.30·32-s − 9.02·34-s + 2.75·35-s + 9.66·37-s + 16.3·38-s − 0.744·40-s + 5.42·41-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.804·4-s + 0.447·5-s + 1.04·7-s − 0.263·8-s + 0.600·10-s + 1.02·11-s + 1.09·13-s + 1.39·14-s − 1.15·16-s − 1.15·17-s + 1.97·19-s + 0.359·20-s + 1.38·22-s + 0.00642·23-s + 0.200·25-s + 1.47·26-s + 0.836·28-s − 1.20·29-s − 0.741·31-s − 1.29·32-s − 1.54·34-s + 0.465·35-s + 1.58·37-s + 2.65·38-s − 0.117·40-s + 0.846·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.206112395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.206112395\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 + 4.75T + 17T^{2} \) |
| 19 | \( 1 - 8.61T + 19T^{2} \) |
| 23 | \( 1 - 0.0307T + 23T^{2} \) |
| 29 | \( 1 + 6.50T + 29T^{2} \) |
| 31 | \( 1 + 4.13T + 31T^{2} \) |
| 37 | \( 1 - 9.66T + 37T^{2} \) |
| 41 | \( 1 - 5.42T + 41T^{2} \) |
| 43 | \( 1 + 2.68T + 43T^{2} \) |
| 47 | \( 1 + 6.91T + 47T^{2} \) |
| 53 | \( 1 - 6.20T + 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 - 0.746T + 61T^{2} \) |
| 67 | \( 1 + 3.61T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 6.97T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.501242541140240673362549608441, −7.50801402068838320150982886853, −6.75093820909228633966887624486, −5.94449309303648542813174329896, −5.46430485291633649953424800419, −4.63318565888540349388915439342, −3.97806820791964976771908740177, −3.24635424138931719728924753261, −2.12170094343344516553379351692, −1.19451736646277212759612891173,
1.19451736646277212759612891173, 2.12170094343344516553379351692, 3.24635424138931719728924753261, 3.97806820791964976771908740177, 4.63318565888540349388915439342, 5.46430485291633649953424800419, 5.94449309303648542813174329896, 6.75093820909228633966887624486, 7.50801402068838320150982886853, 8.501242541140240673362549608441
