| L(s) = 1 | + (−1.68 + 1.08i)2-s + (0.625 − 4.34i)3-s + (1.66 − 3.63i)4-s + (−12.8 + 3.76i)5-s + (3.65 + 7.99i)6-s + (−19.0 − 21.9i)7-s + (1.13 + 7.91i)8-s + (7.37 + 2.16i)9-s + (17.5 − 20.1i)10-s + (−48.5 − 31.1i)11-s + (−14.7 − 9.50i)12-s + (34.5 − 39.8i)13-s + (55.8 + 16.3i)14-s + (8.35 + 58.1i)15-s + (−10.4 − 12.0i)16-s + (27.5 + 60.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (0.120 − 0.837i)3-s + (0.207 − 0.454i)4-s + (−1.14 + 0.336i)5-s + (0.248 + 0.544i)6-s + (−1.02 − 1.18i)7-s + (0.0503 + 0.349i)8-s + (0.273 + 0.0802i)9-s + (0.553 − 0.638i)10-s + (−1.32 − 0.854i)11-s + (−0.355 − 0.228i)12-s + (0.737 − 0.850i)13-s + (1.06 + 0.312i)14-s + (0.143 + 1.00i)15-s + (−0.163 − 0.188i)16-s + (0.393 + 0.861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.224374 - 0.464678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.224374 - 0.464678i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.68 - 1.08i)T \) |
| 23 | \( 1 + (-108. - 22.1i)T \) |
| good | 3 | \( 1 + (-0.625 + 4.34i)T + (-25.9 - 7.60i)T^{2} \) |
| 5 | \( 1 + (12.8 - 3.76i)T + (105. - 67.5i)T^{2} \) |
| 7 | \( 1 + (19.0 + 21.9i)T + (-48.8 + 339. i)T^{2} \) |
| 11 | \( 1 + (48.5 + 31.1i)T + (552. + 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-34.5 + 39.8i)T + (-312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-27.5 - 60.3i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (28.3 - 61.9i)T + (-4.49e3 - 5.18e3i)T^{2} \) |
| 29 | \( 1 + (25.6 + 56.1i)T + (-1.59e4 + 1.84e4i)T^{2} \) |
| 31 | \( 1 + (43.9 + 305. i)T + (-2.85e4 + 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-98.2 - 28.8i)T + (4.26e4 + 2.73e4i)T^{2} \) |
| 41 | \( 1 + (109. - 32.2i)T + (5.79e4 - 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-20.4 + 142. i)T + (-7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 + 76.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + (216. + 250. i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (188. - 217. i)T + (-2.92e4 - 2.03e5i)T^{2} \) |
| 61 | \( 1 + (106. + 737. i)T + (-2.17e5 + 6.39e4i)T^{2} \) |
| 67 | \( 1 + (374. - 240. i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-433. + 278. i)T + (1.48e5 - 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-107. + 235. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (58.3 - 67.3i)T + (-7.01e4 - 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-447. - 131. i)T + (4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-19.0 + 132. i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-653. + 191. i)T + (7.67e5 - 4.93e5i)T^{2} \) |
| 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
−15.20579492925214433485473000371, −13.46949394666293912630907774468, −12.84730724429158031963299489076, −11.02541672370251983947875001248, −10.19123163510529862383425968389, −8.032230872115009996620910420192, −7.59609579042064665091980412801, −6.21713006822763656608123051093, −3.53393460243664936197408253864, −0.47844155274536141927416387379,
3.09322821971293700795321897812, 4.76803930739601986210807839220, 7.10149118354089150962688929107, 8.714950969462466998613597100669, 9.525910378906517496063911002654, 10.81965903363044009439245724115, 12.12834959905847668302091981343, 12.93255098013045143701594960256, 15.17568029841813848510429718531, 15.87777311195208861764783955947
