| L(s) = 1 | − 2i·3-s + (−1 + 2i)5-s − 9-s − 2i·11-s + (3 − 2i)13-s + (4 + 2i)15-s + 6i·19-s − 6i·23-s + (−3 − 4i)25-s − 4i·27-s + 6·29-s − 6i·31-s − 4·33-s + 6·37-s + (−4 − 6i)39-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (−0.447 + 0.894i)5-s − 0.333·9-s − 0.603i·11-s + (0.832 − 0.554i)13-s + (1.03 + 0.516i)15-s + 1.37i·19-s − 1.25i·23-s + (−0.600 − 0.800i)25-s − 0.769i·27-s + 1.11·29-s − 1.07i·31-s − 0.696·33-s + 0.986·37-s + (−0.640 − 0.960i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.464302170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464302170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 2iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 8iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.943309187719180617495087199600, −8.464388885528411953800306968009, −8.051035329127673152906327712165, −7.25089135135114879183420652461, −6.31426835738730583367383529068, −5.93797241021902873688716389666, −4.26546857643922934173021688713, −3.25963735609721207630069688768, −2.20106980713180414349391187764, −0.76070552880741975672395170803,
1.32405409542213580406814219862, 3.09176903652393932978968905610, 4.21884196437809164034625151005, 4.63330733158840173947606055483, 5.53993655651831529876263753437, 6.78552918420920999589288777055, 7.72557261735984306194659785749, 8.835489011682261773493672843094, 9.185556359148498521764584426874, 10.00173840385083004115063785330
