| L(s) = 1 | + (−0.761 − 2.34i)2-s + (0.433 + 0.314i)3-s + (1.55 − 1.13i)4-s + (−0.474 + 1.46i)5-s + (0.408 − 1.25i)6-s + (−22.8 + 16.6i)7-s + (−19.7 − 14.3i)8-s + (−8.25 − 25.4i)9-s + 3.78·10-s + 1.03·12-s + (−21.1 − 65.1i)13-s + (56.3 + 40.9i)14-s + (−0.665 + 0.483i)15-s + (−13.8 + 42.6i)16-s + (−17.1 + 52.6i)17-s + (−53.2 + 38.6i)18-s + ⋯ |
| L(s) = 1 | + (−0.269 − 0.828i)2-s + (0.0834 + 0.0606i)3-s + (0.194 − 0.141i)4-s + (−0.0424 + 0.130i)5-s + (0.0277 − 0.0854i)6-s + (−1.23 + 0.896i)7-s + (−0.874 − 0.635i)8-s + (−0.305 − 0.940i)9-s + 0.119·10-s + 0.0248·12-s + (−0.451 − 1.38i)13-s + (1.07 + 0.781i)14-s + (−0.0114 + 0.00832i)15-s + (−0.216 + 0.666i)16-s + (−0.244 + 0.751i)17-s + (−0.697 + 0.506i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0815826 + 0.554475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0815826 + 0.554475i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (0.761 + 2.34i)T + (-6.47 + 4.70i)T^{2} \) |
| 3 | \( 1 + (-0.433 - 0.314i)T + (8.34 + 25.6i)T^{2} \) |
| 5 | \( 1 + (0.474 - 1.46i)T + (-101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (22.8 - 16.6i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (21.1 + 65.1i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (17.1 - 52.6i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (44.6 + 32.4i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-91.5 + 66.5i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-21.8 - 67.3i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-170. + 123. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-155. - 112. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (414. + 301. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (116. + 357. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-409. + 297. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-144. + 445. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + 289.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (121. - 374. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-234. + 170. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-52.4 - 161. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-93.7 + 288. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-198. - 610. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−12.50217242995243905454381077284, −11.39382680646558148919050141618, −10.19005949327394831380099026019, −9.560990053568067151867126986338, −8.443881645237445763159488330434, −6.57346796801070070292206133568, −5.83858175122522227897629501526, −3.48784909604929572823750539533, −2.52481244496077353377546368849, −0.28896211135167310367922754067,
2.58711210525798881675510214889, 4.36535884224654640842107614210, 6.15251347625967940821750094462, 6.98945487304161422517170375625, 7.945753222452431812282053532432, 9.146702962920107682299437071927, 10.26034588694508625068040233124, 11.52091026638325544394200074324, 12.57717594272130913609340832804, 13.78786699936942081498449546677
