| L(s) = 1 | − 5-s + 11-s − 2·13-s − 6·17-s − 8·19-s + 6·23-s + 25-s − 6·29-s − 2·31-s + 2·37-s + 8·43-s − 12·47-s − 6·53-s − 55-s + 6·59-s − 8·61-s + 2·65-s + 2·67-s + 10·73-s + 8·79-s + 12·83-s + 6·85-s + 6·89-s + 8·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s + 0.328·37-s + 1.21·43-s − 1.75·47-s − 0.824·53-s − 0.134·55-s + 0.781·59-s − 1.02·61-s + 0.248·65-s + 0.244·67-s + 1.17·73-s + 0.900·79-s + 1.31·83-s + 0.650·85-s + 0.635·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.10754296544320, −13.38132622088177, −12.95856864554648, −12.69827961871305, −12.16205659861651, −11.38262468030053, −11.04967875178996, −10.86265826958120, −10.09841508638417, −9.473166686915669, −8.941874803770184, −8.738643902403676, −7.974680881825163, −7.515293740390228, −6.943295659506878, −6.386364064421590, −6.131969099774426, −5.027216041887357, −4.843477110860442, −4.146401534757653, −3.691966904464821, −2.944763055231988, −2.191909373671666, −1.836817580795059, −0.7049085794792177, 0,
0.7049085794792177, 1.836817580795059, 2.191909373671666, 2.944763055231988, 3.691966904464821, 4.146401534757653, 4.843477110860442, 5.027216041887357, 6.131969099774426, 6.386364064421590, 6.943295659506878, 7.515293740390228, 7.974680881825163, 8.738643902403676, 8.941874803770184, 9.473166686915669, 10.09841508638417, 10.86265826958120, 11.04967875178996, 11.38262468030053, 12.16205659861651, 12.69827961871305, 12.95856864554648, 13.38132622088177, 14.10754296544320
