| L(s) = 1 | − 0.615·2-s + 3-s − 1.62·4-s − 3.91·5-s − 0.615·6-s + 4.24·7-s + 2.22·8-s + 9-s + 2.41·10-s − 3.47·11-s − 1.62·12-s − 1.18·13-s − 2.61·14-s − 3.91·15-s + 1.86·16-s + 17-s − 0.615·18-s − 0.900·19-s + 6.34·20-s + 4.24·21-s + 2.13·22-s + 3.44·23-s + 2.22·24-s + 10.3·25-s + 0.731·26-s + 27-s − 6.87·28-s + ⋯ |
| L(s) = 1 | − 0.435·2-s + 0.577·3-s − 0.810·4-s − 1.75·5-s − 0.251·6-s + 1.60·7-s + 0.788·8-s + 0.333·9-s + 0.762·10-s − 1.04·11-s − 0.467·12-s − 0.329·13-s − 0.698·14-s − 1.01·15-s + 0.467·16-s + 0.242·17-s − 0.145·18-s − 0.206·19-s + 1.41·20-s + 0.925·21-s + 0.455·22-s + 0.718·23-s + 0.455·24-s + 2.06·25-s + 0.143·26-s + 0.192·27-s − 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 + 0.615T + 2T^{2} \) |
| 5 | \( 1 + 3.91T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 19 | \( 1 + 0.900T + 19T^{2} \) |
| 23 | \( 1 - 3.44T + 23T^{2} \) |
| 29 | \( 1 + 9.27T + 29T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 - 5.94T + 41T^{2} \) |
| 43 | \( 1 - 2.86T + 43T^{2} \) |
| 47 | \( 1 - 3.86T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 + 3.45T + 59T^{2} \) |
| 61 | \( 1 - 8.01T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 8.74T + 71T^{2} \) |
| 73 | \( 1 - 4.01T + 73T^{2} \) |
| 83 | \( 1 - 5.97T + 83T^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 - 6.09T + 97T^{2} \) |
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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
−7.960459132389410229637781644795, −7.53795123203983647719751231201, −7.40429393107542464014077215614, −5.50313964776287678236662215973, −4.88910714524200958390716719167, −4.22190256229650906859519272736, −3.64521199136164145290750147861, −2.44996870656012271312450261744, −1.20278297213740556529666081362, 0,
1.20278297213740556529666081362, 2.44996870656012271312450261744, 3.64521199136164145290750147861, 4.22190256229650906859519272736, 4.88910714524200958390716719167, 5.50313964776287678236662215973, 7.40429393107542464014077215614, 7.53795123203983647719751231201, 7.960459132389410229637781644795
