L(s) = 1 | − 5.95i·5-s + 12.0i·7-s − 22.0·11-s − 57.4·13-s + 97.0i·17-s − 91.8i·19-s + 137.·23-s + 89.5·25-s − 29.4i·29-s − 149. i·31-s + 71.8·35-s + 216.·37-s − 157. i·41-s + 58.4i·43-s − 196.·47-s + ⋯ |
L(s) = 1 | − 0.532i·5-s + 0.651i·7-s − 0.603·11-s − 1.22·13-s + 1.38i·17-s − 1.10i·19-s + 1.25·23-s + 0.716·25-s − 0.188i·29-s − 0.863i·31-s + 0.346·35-s + 0.962·37-s − 0.599i·41-s + 0.207i·43-s − 0.610·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.597865065\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.597865065\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 5.95iT - 125T^{2} \) |
| 7 | \( 1 - 12.0iT - 343T^{2} \) |
| 11 | \( 1 + 22.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 57.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 97.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 91.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 29.4iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 149. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 216.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 157. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 58.4iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 196.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 35.9iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 316.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 7.87T + 2.26e5T^{2} \) |
| 67 | \( 1 + 424. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 350.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 678.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.08e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 3.90T + 5.71e5T^{2} \) |
| 89 | \( 1 + 156. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.57e3T + 9.12e5T^{2} \) |
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
−9.515695689375916027263349412383, −8.900395185295743447154771305444, −8.044243657136412170522225881619, −7.17344622578332541986616903854, −6.11085832989377218873994591713, −5.14430030690874005415751974184, −4.50913199813616604665044962080, −3.01896823090547902289455438867, −2.08308891095289542822753199907, −0.54013399187183890066888150298,
0.874833494702933364919948148327, 2.48973561588671924985994414538, 3.29350084905611022598980140035, 4.65047848377114237385064800686, 5.32405128675060109486605141019, 6.69008650212509478768199075303, 7.26761249560866754667387642679, 7.991817882820663155321554152778, 9.196506804466302273530356530544, 9.966008059365970414155776949962