| L(s) = 1 | + 1.50·2-s − 0.350·3-s + 0.265·4-s + 3.19·5-s − 0.527·6-s − 2.61·8-s − 2.87·9-s + 4.81·10-s + 0.734·11-s − 0.0929·12-s − 6.80·13-s − 1.12·15-s − 4.45·16-s − 1.13·17-s − 4.32·18-s − 7.92·19-s + 0.847·20-s + 1.10·22-s − 4.32·23-s + 0.915·24-s + 5.22·25-s − 10.2·26-s + 2.06·27-s + 3.06·29-s − 1.68·30-s − 2.67·31-s − 1.48·32-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.202·3-s + 0.132·4-s + 1.42·5-s − 0.215·6-s − 0.923·8-s − 0.959·9-s + 1.52·10-s + 0.221·11-s − 0.0268·12-s − 1.88·13-s − 0.289·15-s − 1.11·16-s − 0.274·17-s − 1.02·18-s − 1.81·19-s + 0.189·20-s + 0.235·22-s − 0.902·23-s + 0.186·24-s + 1.04·25-s − 2.00·26-s + 0.396·27-s + 0.569·29-s − 0.308·30-s − 0.479·31-s − 0.263·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.50T + 2T^{2} \) |
| 3 | \( 1 + 0.350T + 3T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 11 | \( 1 - 0.734T + 11T^{2} \) |
| 13 | \( 1 + 6.80T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 23 | \( 1 + 4.32T + 23T^{2} \) |
| 29 | \( 1 - 3.06T + 29T^{2} \) |
| 31 | \( 1 + 2.67T + 31T^{2} \) |
| 37 | \( 1 - 1.08T + 37T^{2} \) |
| 43 | \( 1 - 2.52T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 + 8.08T + 59T^{2} \) |
| 61 | \( 1 + 3.64T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 - 6.31T + 73T^{2} \) |
| 79 | \( 1 - 4.34T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.962423586396929760862840020896, −8.025285585785388647684909662787, −6.65667035750604793334878170217, −6.24300219277520770226948500390, −5.42563846529821004329376901820, −4.89906889942752140791791289656, −3.97408203852867801788684095110, −2.59876128913809579364148621599, −2.21261130207056394208407837623, 0,
2.21261130207056394208407837623, 2.59876128913809579364148621599, 3.97408203852867801788684095110, 4.89906889942752140791791289656, 5.42563846529821004329376901820, 6.24300219277520770226948500390, 6.65667035750604793334878170217, 8.025285585785388647684909662787, 8.962423586396929760862840020896
