L(s) = 1 | + (−0.813 + 2.70i)2-s + (2.61 + 2.61i)3-s + (−6.67 − 4.40i)4-s + (−0.435 + 11.1i)5-s + (−9.22 + 4.96i)6-s + (17.7 − 17.7i)7-s + (17.3 − 14.4i)8-s − 13.2i·9-s + (−29.9 − 10.2i)10-s + 7.37i·11-s + (−5.93 − 29.0i)12-s + (−2.68 + 2.68i)13-s + (33.6 + 62.6i)14-s + (−30.3 + 28.1i)15-s + (25.1 + 58.8i)16-s + (−20.2 − 20.2i)17-s + ⋯ |
L(s) = 1 | + (−0.287 + 0.957i)2-s + (0.503 + 0.503i)3-s + (−0.834 − 0.551i)4-s + (−0.0389 + 0.999i)5-s + (−0.627 + 0.337i)6-s + (0.959 − 0.959i)7-s + (0.767 − 0.640i)8-s − 0.492i·9-s + (−0.945 − 0.324i)10-s + 0.202i·11-s + (−0.142 − 0.698i)12-s + (−0.0572 + 0.0572i)13-s + (0.643 + 1.19i)14-s + (−0.523 + 0.483i)15-s + (0.392 + 0.919i)16-s + (−0.288 − 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0689 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0689 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.767720 + 0.716469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.767720 + 0.716469i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.813 - 2.70i)T \) |
| 5 | \( 1 + (0.435 - 11.1i)T \) |
good | 3 | \( 1 + (-2.61 - 2.61i)T + 27iT^{2} \) |
| 7 | \( 1 + (-17.7 + 17.7i)T - 343iT^{2} \) |
| 11 | \( 1 - 7.37iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (2.68 - 2.68i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (20.2 + 20.2i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 + 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-71.0 - 71.0i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 34.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 187. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-250. - 250. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 211.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (46.7 + 46.7i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (189. - 189. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (74.5 - 74.5i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + 101.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-34.7 + 34.7i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 614. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (37.4 - 37.4i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-423. - 423. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.04e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (536. + 536. i)T + 9.12e5iT^{2} \) |
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
−17.89191280119431106756791071822, −16.99465369633861029122404903318, −15.21369577111709536489444345271, −14.74449790639662437803013437598, −13.58156956375976491228886300974, −10.99740939227831827610610328226, −9.735239426817691807917541632119, −8.077323041626304831663517728510, −6.68156544128264921793062924906, −4.24966230655131450597625971537,
2.00358922228275972764457728887, 4.86748403966537872443889530926, 8.213996252559873179681797719489, 8.829687128124095342017662194647, 10.90226513816481719029646969690, 12.32776712472532127904171319281, 13.21251725643605531490933266890, 14.68511718395812670323069746590, 16.63249650174215171300059074201, 17.86881611474240504907332638523