| L(s) = 1 | − 5-s − 2·7-s − 2·11-s + 2·13-s + 8·17-s − 4·19-s + 23-s + 25-s − 2·29-s − 8·31-s + 2·35-s + 8·37-s − 2·41-s − 8·43-s + 8·47-s − 3·49-s + 6·53-s + 2·55-s − 8·59-s − 8·61-s − 2·65-s + 16·67-s + 8·71-s − 10·73-s + 4·77-s − 10·79-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.603·11-s + 0.554·13-s + 1.94·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.338·35-s + 1.31·37-s − 0.312·41-s − 1.21·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.269·55-s − 1.04·59-s − 1.02·61-s − 0.248·65-s + 1.95·67-s + 0.949·71-s − 1.17·73-s + 0.455·77-s − 1.12·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−16.37594366444477, −15.50409729314140, −15.26319858225230, −14.51284402408460, −14.10725245377938, −13.21589575057730, −12.88272111261954, −12.42803257889415, −11.76038167280481, −11.12138196104962, −10.56958559037019, −10.00764180232660, −9.461362236101289, −8.767422505548983, −8.125492229296108, −7.596541381578852, −7.034243888002365, −6.207976223641664, −5.714058772288135, −5.062086956382248, −4.158255222902260, −3.486970500810093, −3.043138833046598, −2.041155999973124, −1.012644342302131, 0,
1.012644342302131, 2.041155999973124, 3.043138833046598, 3.486970500810093, 4.158255222902260, 5.062086956382248, 5.714058772288135, 6.207976223641664, 7.034243888002365, 7.596541381578852, 8.125492229296108, 8.767422505548983, 9.461362236101289, 10.00764180232660, 10.56958559037019, 11.12138196104962, 11.76038167280481, 12.42803257889415, 12.88272111261954, 13.21589575057730, 14.10725245377938, 14.51284402408460, 15.26319858225230, 15.50409729314140, 16.37594366444477
