| L(s) = 1 | − 1.71·2-s + 1.21·3-s + 0.934·4-s − 5-s − 2.08·6-s − 4.29·7-s + 1.82·8-s − 1.51·9-s + 1.71·10-s − 11-s + 1.13·12-s − 3.59·13-s + 7.35·14-s − 1.21·15-s − 4.99·16-s + 5.70·17-s + 2.60·18-s + 2.87·19-s − 0.934·20-s − 5.22·21-s + 1.71·22-s + 5.29·23-s + 2.22·24-s + 25-s + 6.15·26-s − 5.49·27-s − 4.01·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 0.702·3-s + 0.467·4-s − 0.447·5-s − 0.850·6-s − 1.62·7-s + 0.645·8-s − 0.506·9-s + 0.541·10-s − 0.301·11-s + 0.328·12-s − 0.996·13-s + 1.96·14-s − 0.314·15-s − 1.24·16-s + 1.38·17-s + 0.613·18-s + 0.659·19-s − 0.208·20-s − 1.14·21-s + 0.365·22-s + 1.10·23-s + 0.453·24-s + 0.200·25-s + 1.20·26-s − 1.05·27-s − 0.758·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 3 | \( 1 - 1.21T + 3T^{2} \) |
| 7 | \( 1 + 4.29T + 7T^{2} \) |
| 13 | \( 1 + 3.59T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 - 2.87T + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 - 7.75T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 - 5.15T + 37T^{2} \) |
| 41 | \( 1 - 4.71T + 41T^{2} \) |
| 43 | \( 1 + 2.15T + 43T^{2} \) |
| 47 | \( 1 - 8.55T + 47T^{2} \) |
| 53 | \( 1 - 1.11T + 53T^{2} \) |
| 59 | \( 1 - 3.59T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 6.62T + 71T^{2} \) |
| 79 | \( 1 - 0.822T + 79T^{2} \) |
| 83 | \( 1 + 9.59T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.155997945729083155757109013031, −7.48508521633334673049922277842, −7.09407637765086285001627939067, −6.03165838610031318301977559347, −5.13196077249472692421196704896, −4.03916125345788890944939793727, −2.97301435783026965516858054528, −2.73049613012457515337939400769, −1.04607778047591573536164220169, 0,
1.04607778047591573536164220169, 2.73049613012457515337939400769, 2.97301435783026965516858054528, 4.03916125345788890944939793727, 5.13196077249472692421196704896, 6.03165838610031318301977559347, 7.09407637765086285001627939067, 7.48508521633334673049922277842, 8.155997945729083155757109013031
