| L(s) = 1 | + 2.69·2-s + 2.56·3-s + 5.25·4-s − 5-s + 6.89·6-s + 8.76·8-s + 3.56·9-s − 2.69·10-s − 3.38·11-s + 13.4·12-s + 2.46·13-s − 2.56·15-s + 13.1·16-s + 2.64·17-s + 9.59·18-s − 3.38·19-s − 5.25·20-s − 9.12·22-s − 23-s + 22.4·24-s + 25-s + 6.63·26-s + 1.43·27-s + 9.20·29-s − 6.89·30-s + 5.10·31-s + 17.7·32-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 1.47·3-s + 2.62·4-s − 0.447·5-s + 2.81·6-s + 3.09·8-s + 1.18·9-s − 0.851·10-s − 1.02·11-s + 3.88·12-s + 0.683·13-s − 0.661·15-s + 3.27·16-s + 0.641·17-s + 2.26·18-s − 0.777·19-s − 1.17·20-s − 1.94·22-s − 0.208·23-s + 4.58·24-s + 0.200·25-s + 1.30·26-s + 0.276·27-s + 1.70·29-s − 1.25·30-s + 0.917·31-s + 3.14·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.24240981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.24240981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 29 | \( 1 - 9.20T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 - 5.50T + 37T^{2} \) |
| 41 | \( 1 - 1.20T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 + 8.21T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 8.28T + 59T^{2} \) |
| 61 | \( 1 + 0.263T + 61T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 + 0.0150T + 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 + 8.67T + 79T^{2} \) |
| 83 | \( 1 - 3.52T + 83T^{2} \) |
| 89 | \( 1 - 4.66T + 89T^{2} \) |
| 97 | \( 1 - 4.09T + 97T^{2} \) |
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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
−8.049644958520682496461376235252, −7.44664446096790196445631267350, −6.50624365235741616712591521433, −5.97697447529855134413175782829, −4.89413428959400304675312183853, −4.39942293525918347761900472528, −3.62023299081961254241894886246, −2.93043738221903256766281908980, −2.56260388888934804364096441591, −1.46197612531999069169810864172,
1.46197612531999069169810864172, 2.56260388888934804364096441591, 2.93043738221903256766281908980, 3.62023299081961254241894886246, 4.39942293525918347761900472528, 4.89413428959400304675312183853, 5.97697447529855134413175782829, 6.50624365235741616712591521433, 7.44664446096790196445631267350, 8.049644958520682496461376235252
