| L(s) = 1 | + 0.0919·2-s − 1.99·4-s + 5-s − 4.79·7-s − 0.367·8-s + 0.0919·10-s + 6.16·11-s − 2.11·13-s − 0.440·14-s + 3.94·16-s − 2.39·17-s − 4.72·19-s − 1.99·20-s + 0.567·22-s + 5.37·23-s + 25-s − 0.194·26-s + 9.54·28-s + 9.61·29-s − 0.532·31-s + 1.09·32-s − 0.219·34-s − 4.79·35-s + 1.10·37-s − 0.434·38-s − 0.367·40-s − 8.65·41-s + ⋯ |
| L(s) = 1 | + 0.0650·2-s − 0.995·4-s + 0.447·5-s − 1.81·7-s − 0.129·8-s + 0.0290·10-s + 1.85·11-s − 0.587·13-s − 0.117·14-s + 0.987·16-s − 0.579·17-s − 1.08·19-s − 0.445·20-s + 0.120·22-s + 1.12·23-s + 0.200·25-s − 0.0382·26-s + 1.80·28-s + 1.78·29-s − 0.0956·31-s + 0.193·32-s − 0.0377·34-s − 0.810·35-s + 0.181·37-s − 0.0705·38-s − 0.0580·40-s − 1.35·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0919T + 2T^{2} \) |
| 7 | \( 1 + 4.79T + 7T^{2} \) |
| 11 | \( 1 - 6.16T + 11T^{2} \) |
| 13 | \( 1 + 2.11T + 13T^{2} \) |
| 17 | \( 1 + 2.39T + 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 - 9.61T + 29T^{2} \) |
| 31 | \( 1 + 0.532T + 31T^{2} \) |
| 37 | \( 1 - 1.10T + 37T^{2} \) |
| 41 | \( 1 + 8.65T + 41T^{2} \) |
| 43 | \( 1 - 3.89T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 + 8.00T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 0.486T + 71T^{2} \) |
| 73 | \( 1 - 6.00T + 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 + 2.03T + 83T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.415194740218648047124303449884, −7.03427541709660185167249048725, −6.51749025227875830289025443607, −6.10877935650496460806427205189, −4.94615958391764180145787945394, −4.23761208464322623245035494779, −3.48407475717377061615684081933, −2.69031241808681163119133096786, −1.21946459402718844531882967646, 0,
1.21946459402718844531882967646, 2.69031241808681163119133096786, 3.48407475717377061615684081933, 4.23761208464322623245035494779, 4.94615958391764180145787945394, 6.10877935650496460806427205189, 6.51749025227875830289025443607, 7.03427541709660185167249048725, 8.415194740218648047124303449884
