| L(s) = 1 | + 2-s + 1.45·3-s + 4-s + 3.97·5-s + 1.45·6-s − 1.21·7-s + 8-s − 0.889·9-s + 3.97·10-s + 5.76·11-s + 1.45·12-s − 3.05·13-s − 1.21·14-s + 5.76·15-s + 16-s − 4.87·17-s − 0.889·18-s − 19-s + 3.97·20-s − 1.76·21-s + 5.76·22-s + 4.80·23-s + 1.45·24-s + 10.7·25-s − 3.05·26-s − 5.65·27-s − 1.21·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.838·3-s + 0.5·4-s + 1.77·5-s + 0.593·6-s − 0.458·7-s + 0.353·8-s − 0.296·9-s + 1.25·10-s + 1.73·11-s + 0.419·12-s − 0.846·13-s − 0.324·14-s + 1.48·15-s + 0.250·16-s − 1.18·17-s − 0.209·18-s − 0.229·19-s + 0.887·20-s − 0.384·21-s + 1.22·22-s + 1.00·23-s + 0.296·24-s + 2.15·25-s − 0.598·26-s − 1.08·27-s − 0.229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.488763755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.488763755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 1.45T + 3T^{2} \) |
| 5 | \( 1 - 3.97T + 5T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 + 4.87T + 17T^{2} \) |
| 23 | \( 1 - 4.80T + 23T^{2} \) |
| 29 | \( 1 - 5.96T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 2.67T + 37T^{2} \) |
| 41 | \( 1 + 7.24T + 41T^{2} \) |
| 43 | \( 1 - 9.61T + 43T^{2} \) |
| 47 | \( 1 - 6.54T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 0.248T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 0.642T + 67T^{2} \) |
| 71 | \( 1 + 5.96T + 71T^{2} \) |
| 73 | \( 1 + 6.74T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 7.32T + 89T^{2} \) |
| 97 | \( 1 + 5.02T + 97T^{2} \) |
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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
−7.78353643797567312261185895732, −6.68145887041892816288129481064, −6.48447000448233765872104243484, −5.95259130145173176868377402562, −4.87653821515927897459448879867, −4.43106074407745021468092257126, −3.26979113182773924577648316182, −2.68571557536084635231525525033, −2.10035905856866506740079706506, −1.16533827511722347768666403445,
1.16533827511722347768666403445, 2.10035905856866506740079706506, 2.68571557536084635231525525033, 3.26979113182773924577648316182, 4.43106074407745021468092257126, 4.87653821515927897459448879867, 5.95259130145173176868377402562, 6.48447000448233765872104243484, 6.68145887041892816288129481064, 7.78353643797567312261185895732
