L(s) = 1 | + (−2.82 + 2.83i)2-s + (−0.0429 − 15.9i)4-s − 11.7·5-s + 58.3i·7-s + (45.4 + 45.0i)8-s + (33.2 − 33.3i)10-s − 100. i·11-s − 170.·13-s + (−165. − 164. i)14-s + (−255. + 1.37i)16-s + 398.·17-s + 404. i·19-s + (0.506 + 188. i)20-s + (284. + 283. i)22-s + 336. i·23-s + ⋯ |
L(s) = 1 | + (−0.706 + 0.708i)2-s + (−0.00268 − 0.999i)4-s − 0.471·5-s + 1.19i·7-s + (0.709 + 0.704i)8-s + (0.332 − 0.333i)10-s − 0.829i·11-s − 1.00·13-s + (−0.843 − 0.841i)14-s + (−0.999 + 0.00536i)16-s + 1.37·17-s + 1.12i·19-s + (0.00126 + 0.471i)20-s + (0.587 + 0.586i)22-s + 0.635i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00268 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.00268 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2169614586\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2169614586\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.82 - 2.83i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 11.7T + 625T^{2} \) |
| 7 | \( 1 - 58.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 100. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 170.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 398.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 404. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 336. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 655.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 635. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.59e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.46e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.23e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.90e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.29e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.15e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.92e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 3.56e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.63e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.49e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.22e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.15e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 910.T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.76e4T + 8.85e7T^{2} \) |
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
−10.43281869562090574807366284960, −9.733702800463484506851849378990, −8.625081393164248471067583004129, −8.059288104498650537725733033098, −7.02678625197484867477900969411, −5.76148869484217812864887347073, −5.20768935565961040876308716306, −3.37850322052158848166775486409, −1.78247774284188636632558694788, −0.093536226767365081491436004493,
1.12340535259551891111250804177, 2.66915056983477117225727811333, 3.92482333717628935166262903863, 4.83017645484062459915117512520, 6.90805245691473854402183033937, 7.48017788756906224773652467347, 8.338365071770451267997248526752, 9.763608133289875033949926385818, 10.07376172733801163906805780216, 11.14120226293851684863610235885