| L(s) = 1 | + 2.51·2-s + 4.34·4-s − 3.54·5-s + 3.81·7-s + 5.90·8-s − 8.92·10-s + 2.43·11-s − 0.859·13-s + 9.60·14-s + 6.18·16-s + 5.04·17-s − 6.23·19-s − 15.3·20-s + 6.13·22-s + 6.49·23-s + 7.54·25-s − 2.16·26-s + 16.5·28-s − 1.60·29-s + 2.29·31-s + 3.76·32-s + 12.7·34-s − 13.5·35-s − 0.904·37-s − 15.7·38-s − 20.9·40-s + 7.92·41-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.17·4-s − 1.58·5-s + 1.44·7-s + 2.08·8-s − 2.82·10-s + 0.734·11-s − 0.238·13-s + 2.56·14-s + 1.54·16-s + 1.22·17-s − 1.43·19-s − 3.44·20-s + 1.30·22-s + 1.35·23-s + 1.50·25-s − 0.424·26-s + 3.13·28-s − 0.297·29-s + 0.412·31-s + 0.665·32-s + 2.17·34-s − 2.28·35-s − 0.148·37-s − 2.54·38-s − 3.30·40-s + 1.23·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.671587578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.671587578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 + 3.54T + 5T^{2} \) |
| 7 | \( 1 - 3.81T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 + 0.859T + 13T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 + 1.60T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 0.904T + 37T^{2} \) |
| 41 | \( 1 - 7.92T + 41T^{2} \) |
| 43 | \( 1 + 4.50T + 43T^{2} \) |
| 47 | \( 1 - 2.73T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 5.37T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 - 1.24T + 67T^{2} \) |
| 71 | \( 1 + 2.94T + 71T^{2} \) |
| 73 | \( 1 - 2.46T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 7.75T + 83T^{2} \) |
| 89 | \( 1 - 0.331T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.257264122409984045457880499281, −7.35292196402306112048811050445, −7.08874990346350198278707254822, −5.99457724386445164321017681224, −5.17489263308684753574303236881, −4.51224770501323375098846215412, −4.06605040838302354796366932933, −3.35343451785639958081569963597, −2.34256026032640839091122378007, −1.13112072426996596707711151144,
1.13112072426996596707711151144, 2.34256026032640839091122378007, 3.35343451785639958081569963597, 4.06605040838302354796366932933, 4.51224770501323375098846215412, 5.17489263308684753574303236881, 5.99457724386445164321017681224, 7.08874990346350198278707254822, 7.35292196402306112048811050445, 8.257264122409984045457880499281
