| L(s) = 1 | − 0.141·2-s + 1.61·3-s − 1.98·4-s − 5-s − 0.228·6-s − 0.325·7-s + 0.561·8-s − 0.389·9-s + 0.141·10-s + 1.41·11-s − 3.19·12-s + 4.49·13-s + 0.0459·14-s − 1.61·15-s + 3.88·16-s − 4.84·17-s + 0.0549·18-s − 6.36·19-s + 1.98·20-s − 0.526·21-s − 0.199·22-s + 5.54·23-s + 0.907·24-s + 25-s − 0.634·26-s − 5.47·27-s + 0.645·28-s + ⋯ |
| L(s) = 1 | − 0.0998·2-s + 0.932·3-s − 0.990·4-s − 0.447·5-s − 0.0931·6-s − 0.123·7-s + 0.198·8-s − 0.129·9-s + 0.0446·10-s + 0.425·11-s − 0.923·12-s + 1.24·13-s + 0.0122·14-s − 0.417·15-s + 0.970·16-s − 1.17·17-s + 0.0129·18-s − 1.46·19-s + 0.442·20-s − 0.114·21-s − 0.0424·22-s + 1.15·23-s + 0.185·24-s + 0.200·25-s − 0.124·26-s − 1.05·27-s + 0.121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.570227414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570227414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.141T + 2T^{2} \) |
| 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 + 0.325T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 4.49T + 13T^{2} \) |
| 17 | \( 1 + 4.84T + 17T^{2} \) |
| 19 | \( 1 + 6.36T + 19T^{2} \) |
| 23 | \( 1 - 5.54T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 - 6.75T + 37T^{2} \) |
| 41 | \( 1 - 0.0191T + 41T^{2} \) |
| 43 | \( 1 + 3.49T + 43T^{2} \) |
| 47 | \( 1 - 8.41T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 - 3.85T + 61T^{2} \) |
| 67 | \( 1 - 8.82T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 9.57T + 73T^{2} \) |
| 79 | \( 1 + 1.29T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 3.74T + 97T^{2} \) |
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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
−8.986827068840864633082836836186, −8.367286326428899101885750499767, −8.140453810321446597522838343499, −6.78327252947922476681371958490, −6.12604250680617540913368133128, −4.82744144676536981553789883722, −4.12380350018254964776628070478, −3.42651426198343928785881439135, −2.36401490046200611801700281534, −0.825268650845539068753636517806,
0.825268650845539068753636517806, 2.36401490046200611801700281534, 3.42651426198343928785881439135, 4.12380350018254964776628070478, 4.82744144676536981553789883722, 6.12604250680617540913368133128, 6.78327252947922476681371958490, 8.140453810321446597522838343499, 8.367286326428899101885750499767, 8.986827068840864633082836836186
