| L(s) = 1 | + 2·5-s − 4·11-s − 2·13-s − 6·17-s − 4·19-s − 25-s − 2·29-s − 6·37-s + 2·41-s − 4·43-s + 6·53-s − 8·55-s + 12·59-s − 2·61-s − 4·65-s + 4·67-s + 6·73-s + 16·79-s − 12·83-s − 12·85-s − 14·89-s − 8·95-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s − 1/5·25-s − 0.371·29-s − 0.986·37-s + 0.312·41-s − 0.609·43-s + 0.824·53-s − 1.07·55-s + 1.56·59-s − 0.256·61-s − 0.496·65-s + 0.488·67-s + 0.702·73-s + 1.80·79-s − 1.31·83-s − 1.30·85-s − 1.48·89-s − 0.820·95-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.179488499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.179488499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| 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
−15.33349742436450, −14.75279181504296, −13.97121581589010, −13.63252899199964, −13.09326000588961, −12.71838733385223, −12.13122237248615, −11.24905185619319, −10.94966578814747, −10.24112484083776, −9.915929006156571, −9.279622819229019, −8.585268174252412, −8.246682008602344, −7.383842169559718, −6.852834943692082, −6.311922087017140, −5.535779262358941, −5.184610192039276, −4.444581119294671, −3.783377824523824, −2.739149750118499, −2.265362621087070, −1.755438159327247, −0.3894236220457950,
0.3894236220457950, 1.755438159327247, 2.265362621087070, 2.739149750118499, 3.783377824523824, 4.444581119294671, 5.184610192039276, 5.535779262358941, 6.311922087017140, 6.852834943692082, 7.383842169559718, 8.246682008602344, 8.585268174252412, 9.279622819229019, 9.915929006156571, 10.24112484083776, 10.94966578814747, 11.24905185619319, 12.13122237248615, 12.71838733385223, 13.09326000588961, 13.63252899199964, 13.97121581589010, 14.75279181504296, 15.33349742436450
