| L(s) = 1 | − 2-s − 0.510·3-s + 4-s + 2.36·5-s + 0.510·6-s + 2.78·7-s − 8-s − 2.73·9-s − 2.36·10-s + 0.898·11-s − 0.510·12-s + 1.36·13-s − 2.78·14-s − 1.21·15-s + 16-s − 4.57·17-s + 2.73·18-s + 3.53·19-s + 2.36·20-s − 1.42·21-s − 0.898·22-s − 23-s + 0.510·24-s + 0.613·25-s − 1.36·26-s + 2.93·27-s + 2.78·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.295·3-s + 0.5·4-s + 1.05·5-s + 0.208·6-s + 1.05·7-s − 0.353·8-s − 0.912·9-s − 0.749·10-s + 0.270·11-s − 0.147·12-s + 0.377·13-s − 0.745·14-s − 0.312·15-s + 0.250·16-s − 1.11·17-s + 0.645·18-s + 0.811·19-s + 0.529·20-s − 0.310·21-s − 0.191·22-s − 0.208·23-s + 0.104·24-s + 0.122·25-s − 0.267·26-s + 0.564·27-s + 0.527·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 0.510T + 3T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 11 | \( 1 - 0.898T + 11T^{2} \) |
| 13 | \( 1 - 1.36T + 13T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 - 3.53T + 19T^{2} \) |
| 29 | \( 1 + 9.50T + 29T^{2} \) |
| 31 | \( 1 + 7.83T + 31T^{2} \) |
| 37 | \( 1 + 3.08T + 37T^{2} \) |
| 41 | \( 1 - 6.50T + 41T^{2} \) |
| 43 | \( 1 + 2.06T + 43T^{2} \) |
| 47 | \( 1 - 2.55T + 47T^{2} \) |
| 53 | \( 1 + 0.0206T + 53T^{2} \) |
| 59 | \( 1 + 3.00T + 59T^{2} \) |
| 61 | \( 1 + 8.28T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 2.24T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 + 5.25T + 79T^{2} \) |
| 83 | \( 1 + 6.21T + 83T^{2} \) |
| 89 | \( 1 + 2.45T + 89T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
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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
−7.71425597676455431397640566241, −7.16912493516013396197988930231, −6.13852134150440790244735710741, −5.74280455428181969552267539999, −5.10725736066889176512581664009, −4.06947295599066274383509409443, −2.96543162250558087383050718891, −1.97657387752740263827619880533, −1.48421042742276654901730865522, 0,
1.48421042742276654901730865522, 1.97657387752740263827619880533, 2.96543162250558087383050718891, 4.06947295599066274383509409443, 5.10725736066889176512581664009, 5.74280455428181969552267539999, 6.13852134150440790244735710741, 7.16912493516013396197988930231, 7.71425597676455431397640566241
