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 | \( 1 + 1.76e4T + 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14120226293851684863610235885, −10.07376172733801163906805780216, −9.763608133289875033949926385818, −8.338365071770451267997248526752, −7.48017788756906224773652467347, −6.90805245691473854402183033937, −4.83017645484062459915117512520, −3.92482333717628935166262903863, −2.66915056983477117225727811333, −1.12340535259551891111250804177,
0.093536226767365081491436004493, 1.78247774284188636632558694788, 3.37850322052158848166775486409, 5.20768935565961040876308716306, 5.76148869484217812864887347073, 7.02678625197484867477900969411, 8.059288104498650537725733033098, 8.625081393164248471067583004129, 9.733702800463484506851849378990, 10.43281869562090574807366284960