| L(s) = 1 | − 2-s + 2.04·3-s + 4-s − 0.868·5-s − 2.04·6-s − 8-s + 1.17·9-s + 0.868·10-s − 4.52·11-s + 2.04·12-s + 4.81·13-s − 1.77·15-s + 16-s − 0.259·17-s − 1.17·18-s + 1.04·19-s − 0.868·20-s + 4.52·22-s − 7.24·23-s − 2.04·24-s − 4.24·25-s − 4.81·26-s − 3.73·27-s + 29-s + 1.77·30-s − 2.40·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.17·3-s + 0.5·4-s − 0.388·5-s − 0.833·6-s − 0.353·8-s + 0.390·9-s + 0.274·10-s − 1.36·11-s + 0.589·12-s + 1.33·13-s − 0.457·15-s + 0.250·16-s − 0.0630·17-s − 0.276·18-s + 0.239·19-s − 0.194·20-s + 0.964·22-s − 1.51·23-s − 0.416·24-s − 0.849·25-s − 0.943·26-s − 0.718·27-s + 0.185·29-s + 0.323·30-s − 0.432·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 + 0.868T + 5T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 + 0.259T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 31 | \( 1 + 2.40T + 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 - 5.52T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 6.20T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 6.90T + 83T^{2} \) |
| 89 | \( 1 + 1.73T + 89T^{2} \) |
| 97 | \( 1 - 5.00T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.441837660799892357569101764847, −7.83789111089604825488271897563, −7.37992076265521777871913358225, −6.17434065118425768947074006099, −5.49084081000608624732600961470, −4.12021322966568710099582109378, −3.40877578624537398951727641039, −2.55475386304570757246663340896, −1.66497318558662169749401231185, 0,
1.66497318558662169749401231185, 2.55475386304570757246663340896, 3.40877578624537398951727641039, 4.12021322966568710099582109378, 5.49084081000608624732600961470, 6.17434065118425768947074006099, 7.37992076265521777871913358225, 7.83789111089604825488271897563, 8.441837660799892357569101764847
