| L(s) = 1 | + 8.23i·2-s − 11.6i·3-s − 35.8·4-s + 95.8·6-s − 195. i·7-s − 31.9i·8-s + 107.·9-s + 64.5·11-s + 417. i·12-s − 169i·13-s + 1.60e3·14-s − 885.·16-s + 426. i·17-s + 886. i·18-s + 959.·19-s + ⋯ |
| L(s) = 1 | + 1.45i·2-s − 0.746i·3-s − 1.12·4-s + 1.08·6-s − 1.50i·7-s − 0.176i·8-s + 0.442·9-s + 0.160·11-s + 0.836i·12-s − 0.277i·13-s + 2.19·14-s − 0.864·16-s + 0.357i·17-s + 0.644i·18-s + 0.609·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.288071940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288071940\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 2 | \( 1 - 8.23iT - 32T^{2} \) |
| 3 | \( 1 + 11.6iT - 243T^{2} \) |
| 7 | \( 1 + 195. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 64.5T + 1.61e5T^{2} \) |
| 17 | \( 1 - 426. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 959.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 499. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.21e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 6.49e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.56e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 6.29e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 4.03e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.56e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.91e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.45e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.02e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 4.81e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.33e4iT - 8.58e9T^{2} \) |
| 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
−10.50664813505657075611510300830, −9.467936666385442479415893418167, −8.180687562170295818571829472961, −7.43468974872750279001660989422, −7.00195435856139709439648558374, −6.05107153719232471121279053808, −4.83491302887465145350715053809, −3.70807636165936620376841431572, −1.64530475903753231304915022638, −0.33727142883327324018425899725,
1.43612129908459935773676515898, 2.53193075464804179745251566112, 3.54070962695427092005116606982, 4.62198387361713740041410107573, 5.65159343388496543369243151133, 7.12321734301169355654095133457, 8.775002564156947598188992787146, 9.324518292813121001400524569104, 10.06638045767113916921674940028, 10.99224801499722497449852765273
