| L(s) = 1 | + (2.78 + 0.514i)2-s + (7.47 + 2.86i)4-s + (3.43 + 3.43i)5-s + 8.14·7-s + (19.3 + 11.8i)8-s + (7.79 + 11.3i)10-s + (−1.92 + 1.92i)11-s + (9.16 + 9.16i)13-s + (22.6 + 4.19i)14-s + (47.6 + 42.7i)16-s + 28.6i·17-s + (39.9 − 39.9i)19-s + (15.8 + 35.5i)20-s + (−6.33 + 4.35i)22-s + 80.3i·23-s + ⋯ |
| L(s) = 1 | + (0.983 + 0.181i)2-s + (0.933 + 0.357i)4-s + (0.307 + 0.307i)5-s + 0.439·7-s + (0.853 + 0.521i)8-s + (0.246 + 0.358i)10-s + (−0.0526 + 0.0526i)11-s + (0.195 + 0.195i)13-s + (0.432 + 0.0800i)14-s + (0.744 + 0.668i)16-s + 0.409i·17-s + (0.481 − 0.481i)19-s + (0.177 + 0.396i)20-s + (−0.0613 + 0.0422i)22-s + 0.728i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.14633 + 1.00162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.14633 + 1.00162i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.78 - 0.514i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-3.43 - 3.43i)T + 125iT^{2} \) |
| 7 | \( 1 - 8.14T + 343T^{2} \) |
| 11 | \( 1 + (1.92 - 1.92i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (-9.16 - 9.16i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 - 28.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-39.9 + 39.9i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 - 80.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (113. - 113. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + 306. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-47.8 + 47.8i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 349.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (164. + 164. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 - 40.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + (454. + 454. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-245. + 245. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-186. - 186. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (118. - 118. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 414. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 431. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-739. - 739. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 942.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 983.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.94602746916265852239556027695, −11.73497632416512220413876055603, −11.03839627074017720106548978161, −9.820438407717202179176290783596, −8.269303058103733655320681265297, −7.15846946163837039030635880140, −6.04648931245906481158917157667, −4.92311841952110124443646924012, −3.54925396381596073859205600060, −1.98827022973976082497486273201,
1.53404476926484848914078995035, 3.20529739231004781201329244834, 4.70629912162290267201558174917, 5.65114457204247044670942855749, 6.92032341386078892451692086037, 8.183971013683085990507441780972, 9.646049645134890469209445483767, 10.74377087186345373969294189719, 11.68174948858095014416987379686, 12.61633153049527129479552756188
