L(s) = 1 | + 7.65i·2-s − 5.19·3-s − 42.5·4-s − 38.1·5-s − 39.7i·6-s − 35.1·7-s − 203. i·8-s + 27·9-s − 292. i·10-s + 147. i·11-s + 221.·12-s + 17.0i·13-s − 268. i·14-s + 198.·15-s + 875.·16-s − 354.·17-s + ⋯ |
L(s) = 1 | + 1.91i·2-s − 0.577·3-s − 2.66·4-s − 1.52·5-s − 1.10i·6-s − 0.717·7-s − 3.17i·8-s + 0.333·9-s − 2.92i·10-s + 1.22i·11-s + 1.53·12-s + 0.101i·13-s − 1.37i·14-s + 0.881·15-s + 3.41·16-s − 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.09856197769\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09856197769\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (-1.69e3 + 3.04e3i)T \) |
good | 2 | \( 1 - 7.65iT - 16T^{2} \) |
| 5 | \( 1 + 38.1T + 625T^{2} \) |
| 7 | \( 1 + 35.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 147. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 17.0iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 354.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 647.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 862. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 440.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 347. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 972. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.78e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.83e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.28e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.10e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 5.98e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.35e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 1.39e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 1.64e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 8.35e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 5.02e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 5.29e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.40e4iT - 8.85e7T^{2} \) |
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
−12.61418737296219367138143904187, −11.25693255186467423944610836773, −9.801978116220442959265845260457, −8.768035232997542350301463405147, −7.67605860798592702514365438847, −6.99667244642392890141350159047, −6.13440977216860658335002510265, −4.60964706388521358057447568627, −4.04933073161054073073270575226, −0.084843263354554227351636252656,
0.55170050474535165824157072289, 2.66112814720322335908046778829, 3.87486461034463362682434058250, 4.58441340869422324790799367998, 6.41524183538948616930490350483, 8.307094426927261498145488282820, 8.910515088649204105796565362523, 10.50454325939921198763572830223, 10.90764525343147065685912411235, 11.71494236237603389873214149157