| L(s) = 1 | + (−1.30 + 1.51i)2-s + (2.05 − 1.32i)3-s + (−0.569 − 3.95i)4-s + (5.37 + 11.7i)5-s + (−0.695 + 4.83i)6-s + (20.2 − 5.93i)7-s + (6.73 + 4.32i)8-s + (−8.73 + 19.1i)9-s + (−24.8 − 7.29i)10-s + (29.6 + 34.2i)11-s + (−6.39 − 7.38i)12-s + (−24.4 − 7.17i)13-s + (−17.4 + 38.3i)14-s + (26.6 + 17.0i)15-s + (−15.3 + 4.50i)16-s + (16.3 − 113. i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (0.395 − 0.254i)3-s + (−0.0711 − 0.494i)4-s + (0.480 + 1.05i)5-s + (−0.0473 + 0.329i)6-s + (1.09 − 0.320i)7-s + (0.297 + 0.191i)8-s + (−0.323 + 0.708i)9-s + (−0.785 − 0.230i)10-s + (0.813 + 0.938i)11-s + (−0.153 − 0.177i)12-s + (−0.521 − 0.153i)13-s + (−0.334 + 0.731i)14-s + (0.457 + 0.294i)15-s + (−0.239 + 0.0704i)16-s + (0.232 − 1.61i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.20474 + 0.592881i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20474 + 0.592881i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 - 1.51i)T \) |
| 23 | \( 1 + (107. - 23.2i)T \) |
| good | 3 | \( 1 + (-2.05 + 1.32i)T + (11.2 - 24.5i)T^{2} \) |
| 5 | \( 1 + (-5.37 - 11.7i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (-20.2 + 5.93i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (-29.6 - 34.2i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (24.4 + 7.17i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-16.3 + 113. i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (13.1 + 91.2i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (2.82 - 19.6i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (92.0 + 59.1i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (7.57 - 16.5i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (109. + 240. i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (314. - 202. i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 - 439.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-432. + 126. i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-89.3 - 26.2i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (459. + 295. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-265. + 306. i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (174. - 201. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (169. + 1.17e3i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-897. - 263. i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (83.9 - 183. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (1.15e3 - 744. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-520. - 1.14e3i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
| 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
−15.18976899003195875960539774974, −14.31247454537923881818748594264, −13.76293861043390455226633408521, −11.67703450970241855414352548589, −10.53656299985431252856321770489, −9.292056616621154716674860705299, −7.71174278281995932643971038120, −6.92279431453107631954372944405, −4.99340994072947842448677041390, −2.22225005630261622249754389549,
1.55845212166599112441938167301, 3.96039756284164217999304272628, 5.81637929315444062789148234751, 8.323272876601445833346456340217, 8.818672262613941493145052707356, 10.13914304704849857731153709148, 11.67817684717411344114160016144, 12.50584349949592248696543376510, 14.04015274064306929929911779408, 14.93427566326954966488414196002
