| L(s) = 1 | + 0.891·2-s − 2.80·3-s − 1.20·4-s − 2.73·5-s − 2.49·6-s − 3.81·7-s − 2.85·8-s + 4.84·9-s − 2.43·10-s − 3.01·11-s + 3.37·12-s + 1.64·13-s − 3.39·14-s + 7.65·15-s − 0.136·16-s − 5.40·17-s + 4.31·18-s − 19-s + 3.29·20-s + 10.6·21-s − 2.68·22-s + 0.415·23-s + 8.00·24-s + 2.46·25-s + 1.46·26-s − 5.17·27-s + 4.59·28-s + ⋯ |
| L(s) = 1 | + 0.630·2-s − 1.61·3-s − 0.602·4-s − 1.22·5-s − 1.01·6-s − 1.44·7-s − 1.01·8-s + 1.61·9-s − 0.770·10-s − 0.908·11-s + 0.974·12-s + 0.456·13-s − 0.908·14-s + 1.97·15-s − 0.0340·16-s − 1.31·17-s + 1.01·18-s − 0.229·19-s + 0.736·20-s + 2.33·21-s − 0.572·22-s + 0.0866·23-s + 1.63·24-s + 0.492·25-s + 0.287·26-s − 0.995·27-s + 0.868·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 + T \) |
| 317 | \( 1 + T \) |
| good | 2 | \( 1 - 0.891T + 2T^{2} \) |
| 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 + 3.81T + 7T^{2} \) |
| 11 | \( 1 + 3.01T + 11T^{2} \) |
| 13 | \( 1 - 1.64T + 13T^{2} \) |
| 17 | \( 1 + 5.40T + 17T^{2} \) |
| 23 | \( 1 - 0.415T + 23T^{2} \) |
| 29 | \( 1 + 4.39T + 29T^{2} \) |
| 31 | \( 1 - 2.95T + 31T^{2} \) |
| 37 | \( 1 + 6.54T + 37T^{2} \) |
| 41 | \( 1 - 4.19T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 7.03T + 47T^{2} \) |
| 53 | \( 1 - 5.12T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 0.516T + 61T^{2} \) |
| 67 | \( 1 + 4.09T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 1.34T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 + 5.85T + 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.47201988125871201996577012175, −6.73515002944445240715059296742, −6.20315088174752633343975242975, −5.53322837070437301192647970093, −4.86836903787133291195775332268, −4.08666266356387284085734179901, −3.66764982854708184540083341795, −2.60763647950274900916518890358, −0.60930284182418841304220855216, 0,
0.60930284182418841304220855216, 2.60763647950274900916518890358, 3.66764982854708184540083341795, 4.08666266356387284085734179901, 4.86836903787133291195775332268, 5.53322837070437301192647970093, 6.20315088174752633343975242975, 6.73515002944445240715059296742, 7.47201988125871201996577012175
