| L(s) = 1 | − 2.17·2-s + 3-s + 2.75·4-s + 1.63·5-s − 2.17·6-s + 4.53·7-s − 1.63·8-s + 9-s − 3.55·10-s − 1.15·11-s + 2.75·12-s + 6.10·13-s − 9.88·14-s + 1.63·15-s − 1.93·16-s + 3.44·17-s − 2.17·18-s + 0.492·19-s + 4.48·20-s + 4.53·21-s + 2.51·22-s + 4.54·23-s − 1.63·24-s − 2.34·25-s − 13.3·26-s + 27-s + 12.4·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 0.577·3-s + 1.37·4-s + 0.729·5-s − 0.889·6-s + 1.71·7-s − 0.579·8-s + 0.333·9-s − 1.12·10-s − 0.348·11-s + 0.794·12-s + 1.69·13-s − 2.64·14-s + 0.421·15-s − 0.483·16-s + 0.835·17-s − 0.513·18-s + 0.113·19-s + 1.00·20-s + 0.989·21-s + 0.536·22-s + 0.948·23-s − 0.334·24-s − 0.468·25-s − 2.60·26-s + 0.192·27-s + 2.35·28-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
| \(L(1)\) |
\(\approx\) |
\(2.099874518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.099874518\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.17T + 2T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 7 | \( 1 - 4.53T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 - 6.10T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 - 0.492T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 + 1.04T + 31T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 + 4.05T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 + 4.35T + 53T^{2} \) |
| 59 | \( 1 + 2.85T + 59T^{2} \) |
| 61 | \( 1 - 0.275T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 2.51T + 71T^{2} \) |
| 73 | \( 1 + 3.11T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 8.00T + 83T^{2} \) |
| 89 | \( 1 + 1.95T + 89T^{2} \) |
| 97 | \( 1 - 3.14T + 97T^{2} \) |
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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
−8.133404968839711705057425854980, −7.54593809538958401143004369055, −6.75671744046800998230834024583, −5.83415568949296587153814841426, −5.17862565216288355709018835072, −4.25587961534724818590696870464, −3.25917167281949165399257687441, −2.15509898873276788975732032481, −1.57147887297730646975626015218, −1.00274878708714822272579916252,
1.00274878708714822272579916252, 1.57147887297730646975626015218, 2.15509898873276788975732032481, 3.25917167281949165399257687441, 4.25587961534724818590696870464, 5.17862565216288355709018835072, 5.83415568949296587153814841426, 6.75671744046800998230834024583, 7.54593809538958401143004369055, 8.133404968839711705057425854980
