| L(s) = 1 | + 1.67·2-s − 3-s + 0.806·4-s − 3.48·5-s − 1.67·6-s + 7-s − 1.99·8-s + 9-s − 5.83·10-s + 1.61·11-s − 0.806·12-s + 3.19·13-s + 1.67·14-s + 3.48·15-s − 4.96·16-s + 17-s + 1.67·18-s − 0.193·19-s − 2.80·20-s − 21-s + 2.70·22-s + 5.63·23-s + 1.99·24-s + 7.11·25-s + 5.35·26-s − 27-s + 0.806·28-s + ⋯ |
| L(s) = 1 | + 1.18·2-s − 0.577·3-s + 0.403·4-s − 1.55·5-s − 0.683·6-s + 0.377·7-s − 0.707·8-s + 0.333·9-s − 1.84·10-s + 0.486·11-s − 0.232·12-s + 0.885·13-s + 0.447·14-s + 0.898·15-s − 1.24·16-s + 0.242·17-s + 0.394·18-s − 0.0444·19-s − 0.627·20-s − 0.218·21-s + 0.575·22-s + 1.17·23-s + 0.408·24-s + 1.42·25-s + 1.04·26-s − 0.192·27-s + 0.152·28-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 | \( 1 - 1.67T + 2T^{2} \) |
| 5 | \( 1 + 3.48T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 19 | \( 1 + 0.193T + 19T^{2} \) |
| 23 | \( 1 - 5.63T + 23T^{2} \) |
| 29 | \( 1 - 1.61T + 29T^{2} \) |
| 31 | \( 1 + 5.61T + 31T^{2} \) |
| 37 | \( 1 + 9.96T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 - 4.67T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 2.28T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 + 6.93T + 67T^{2} \) |
| 71 | \( 1 + 0.712T + 71T^{2} \) |
| 73 | \( 1 + 9.83T + 73T^{2} \) |
| 83 | \( 1 + 0.249T + 83T^{2} \) |
| 89 | \( 1 - 5.08T + 89T^{2} \) |
| 97 | \( 1 + 2.86T + 97T^{2} \) |
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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
−7.952175864828672040747777661346, −7.13909275145762287307098906785, −6.53358364011601964055851854694, −5.62722553166650735228088724206, −4.95112563238974988669962134952, −4.24206883561560947188951861024, −3.69546382270552799196243784535, −3.00717991254265907063111670918, −1.33617199897180113315049314583, 0,
1.33617199897180113315049314583, 3.00717991254265907063111670918, 3.69546382270552799196243784535, 4.24206883561560947188951861024, 4.95112563238974988669962134952, 5.62722553166650735228088724206, 6.53358364011601964055851854694, 7.13909275145762287307098906785, 7.952175864828672040747777661346
