| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s + 2·13-s − 14-s + 15-s + 16-s − 2·17-s − 18-s + 4·19-s − 20-s − 21-s − 4·22-s + 8·23-s + 24-s + 25-s − 2·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.206079789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.206079789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.62473579899603, −12.06163373329245, −11.64928990714852, −11.33074121269605, −10.94364049893605, −10.64624767910943, −9.822753502474473, −9.521053609478864, −8.964397196673473, −8.741967071450793, −8.003575715202286, −7.661889774413481, −7.060114043470596, −6.686236026865324, −6.331608764254908, −5.562644412484552, −5.281168153161004, −4.451395157918756, −4.171065610323170, −3.471947629511198, −2.966061653896098, −2.233603682426156, −1.467627568403914, −1.003201136659072, −0.5791902407806221,
0.5791902407806221, 1.003201136659072, 1.467627568403914, 2.233603682426156, 2.966061653896098, 3.471947629511198, 4.171065610323170, 4.451395157918756, 5.281168153161004, 5.562644412484552, 6.331608764254908, 6.686236026865324, 7.060114043470596, 7.661889774413481, 8.003575715202286, 8.741967071450793, 8.964397196673473, 9.521053609478864, 9.822753502474473, 10.64624767910943, 10.94364049893605, 11.33074121269605, 11.64928990714852, 12.06163373329245, 12.62473579899603
