| L(s) = 1 | + 4.99·2-s + 0.616·3-s + 16.9·4-s − 5·5-s + 3.07·6-s − 9.72·7-s + 44.5·8-s − 26.6·9-s − 24.9·10-s + 11·11-s + 10.4·12-s − 60.7·13-s − 48.5·14-s − 3.08·15-s + 86.8·16-s + 64.6·17-s − 132.·18-s − 19·19-s − 84.5·20-s − 5.99·21-s + 54.9·22-s − 170.·23-s + 27.4·24-s + 25·25-s − 303.·26-s − 33.0·27-s − 164.·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.118·3-s + 2.11·4-s − 0.447·5-s + 0.209·6-s − 0.525·7-s + 1.96·8-s − 0.985·9-s − 0.789·10-s + 0.301·11-s + 0.250·12-s − 1.29·13-s − 0.926·14-s − 0.0530·15-s + 1.35·16-s + 0.923·17-s − 1.73·18-s − 0.229·19-s − 0.945·20-s − 0.0622·21-s + 0.532·22-s − 1.54·23-s + 0.233·24-s + 0.200·25-s − 2.28·26-s − 0.235·27-s − 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 4.99T + 8T^{2} \) |
| 3 | \( 1 - 0.616T + 27T^{2} \) |
| 7 | \( 1 + 9.72T + 343T^{2} \) |
| 13 | \( 1 + 60.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 64.6T + 4.91e3T^{2} \) |
| 23 | \( 1 + 170.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 48.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 297.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 42.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 288.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 240.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 339.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 362.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 704.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 506.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 103.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 301.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 446.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 720.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 373.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 6.54T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−9.191591290549337821484718297495, −7.933361911161106027452441726076, −7.25424888887294096668795180572, −6.21902694751155481748257136900, −5.61598353453517833105611827779, −4.69501607167547004253715244227, −3.74487934819123089620748782936, −3.04089312224911360898236024106, −2.08355276861702128535863043858, 0,
2.08355276861702128535863043858, 3.04089312224911360898236024106, 3.74487934819123089620748782936, 4.69501607167547004253715244227, 5.61598353453517833105611827779, 6.21902694751155481748257136900, 7.25424888887294096668795180572, 7.933361911161106027452441726076, 9.191591290549337821484718297495
