| L(s) = 1 | + 0.0962·2-s + 3-s − 1.99·4-s + 5-s + 0.0962·6-s − 2.61·7-s − 0.384·8-s + 9-s + 0.0962·10-s + 1.60·11-s − 1.99·12-s + 5.20·13-s − 0.251·14-s + 15-s + 3.94·16-s + 6.49·17-s + 0.0962·18-s + 2.92·19-s − 1.99·20-s − 2.61·21-s + 0.154·22-s − 1.34·23-s − 0.384·24-s + 25-s + 0.501·26-s + 27-s + 5.20·28-s + ⋯ |
| L(s) = 1 | + 0.0680·2-s + 0.577·3-s − 0.995·4-s + 0.447·5-s + 0.0393·6-s − 0.988·7-s − 0.135·8-s + 0.333·9-s + 0.0304·10-s + 0.485·11-s − 0.574·12-s + 1.44·13-s − 0.0672·14-s + 0.258·15-s + 0.986·16-s + 1.57·17-s + 0.0226·18-s + 0.671·19-s − 0.445·20-s − 0.570·21-s + 0.0330·22-s − 0.280·23-s − 0.0784·24-s + 0.200·25-s + 0.0983·26-s + 0.192·27-s + 0.983·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.239810215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.239810215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0962T + 2T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 13 | \( 1 - 5.20T + 13T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 + 1.34T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 7.73T + 31T^{2} \) |
| 37 | \( 1 - 4.28T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 - 0.403T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 1.08T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 - 1.72T + 71T^{2} \) |
| 73 | \( 1 - 9.56T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 5.02T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.012612823095105952709782549779, −7.62968003730710366201464857300, −6.55395198009292294908981717509, −5.78978387390623831566947860506, −5.43330263887400606294150623240, −4.16212993212790745579775049730, −3.55511972848710871039143060755, −3.17541482817662791980913977820, −1.73332486212095021023370599757, −0.804484538844829513997700710029,
0.804484538844829513997700710029, 1.73332486212095021023370599757, 3.17541482817662791980913977820, 3.55511972848710871039143060755, 4.16212993212790745579775049730, 5.43330263887400606294150623240, 5.78978387390623831566947860506, 6.55395198009292294908981717509, 7.62968003730710366201464857300, 8.012612823095105952709782549779
