| L(s) = 1 | + 3-s − 2.99·5-s − 4.65·7-s + 9-s + 0.962·11-s + 0.976·13-s − 2.99·15-s + 2.56·17-s + 8.42·19-s − 4.65·21-s + 6.29·23-s + 3.96·25-s + 27-s − 3.04·29-s − 10.1·31-s + 0.962·33-s + 13.9·35-s + 10.7·37-s + 0.976·39-s + 0.413·41-s − 5.35·43-s − 2.99·45-s − 8.47·47-s + 14.7·49-s + 2.56·51-s − 13.1·53-s − 2.88·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.33·5-s − 1.76·7-s + 0.333·9-s + 0.290·11-s + 0.270·13-s − 0.773·15-s + 0.621·17-s + 1.93·19-s − 1.01·21-s + 1.31·23-s + 0.792·25-s + 0.192·27-s − 0.565·29-s − 1.82·31-s + 0.167·33-s + 2.35·35-s + 1.76·37-s + 0.156·39-s + 0.0645·41-s − 0.816·43-s − 0.446·45-s − 1.23·47-s + 2.10·49-s + 0.358·51-s − 1.80·53-s − 0.388·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2.99T + 5T^{2} \) |
| 7 | \( 1 + 4.65T + 7T^{2} \) |
| 11 | \( 1 - 0.962T + 11T^{2} \) |
| 13 | \( 1 - 0.976T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 8.42T + 19T^{2} \) |
| 23 | \( 1 - 6.29T + 23T^{2} \) |
| 29 | \( 1 + 3.04T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 0.413T + 41T^{2} \) |
| 43 | \( 1 + 5.35T + 43T^{2} \) |
| 47 | \( 1 + 8.47T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 0.101T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 1.47T + 79T^{2} \) |
| 83 | \( 1 + 4.56T + 83T^{2} \) |
| 89 | \( 1 - 8.20T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.78404155726340048331700017292, −7.55583005159352344158407145798, −6.79248659241408916929437875200, −5.97786728346179991070893941713, −4.98741262851435444414738527203, −3.92887789007512121599471926072, −3.19355338488292817630320468877, −3.11930735800683632948912094801, −1.25977988748857732524129212862, 0,
1.25977988748857732524129212862, 3.11930735800683632948912094801, 3.19355338488292817630320468877, 3.92887789007512121599471926072, 4.98741262851435444414738527203, 5.97786728346179991070893941713, 6.79248659241408916929437875200, 7.55583005159352344158407145798, 7.78404155726340048331700017292
