| L(s) = 1 | + (4.35 + 0.768i)2-s + (3.37 + 1.22i)4-s + (−27.2 − 32.4i)5-s + (−49.1 + 17.8i)7-s + (−47.5 − 27.4i)8-s + (−93.8 − 162. i)10-s + (−11.4 + 13.6i)11-s + (21.2 + 120. i)13-s + (−227. + 40.1i)14-s + (−230. − 193. i)16-s + (377. − 217. i)17-s + (65.4 − 113. i)19-s + (−52.1 − 143. i)20-s + (−60.3 + 50.6i)22-s + (240. − 660. i)23-s + ⋯ |
| L(s) = 1 | + (1.08 + 0.192i)2-s + (0.211 + 0.0768i)4-s + (−1.08 − 1.29i)5-s + (−1.00 + 0.364i)7-s + (−0.743 − 0.429i)8-s + (−0.938 − 1.62i)10-s + (−0.0945 + 0.112i)11-s + (0.125 + 0.713i)13-s + (−1.16 + 0.204i)14-s + (−0.899 − 0.754i)16-s + (1.30 − 0.753i)17-s + (0.181 − 0.313i)19-s + (−0.130 − 0.357i)20-s + (−0.124 + 0.104i)22-s + (0.454 − 1.24i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.333116 - 0.914465i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.333116 - 0.914465i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-4.35 - 0.768i)T + (15.0 + 5.47i)T^{2} \) |
| 5 | \( 1 + (27.2 + 32.4i)T + (-108. + 615. i)T^{2} \) |
| 7 | \( 1 + (49.1 - 17.8i)T + (1.83e3 - 1.54e3i)T^{2} \) |
| 11 | \( 1 + (11.4 - 13.6i)T + (-2.54e3 - 1.44e4i)T^{2} \) |
| 13 | \( 1 + (-21.2 - 120. i)T + (-2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-377. + 217. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-65.4 + 113. i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-240. + 660. i)T + (-2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (809. + 142. i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (367. + 133. i)T + (7.07e5 + 5.93e5i)T^{2} \) |
| 37 | \( 1 + (905. + 1.56e3i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (1.77e3 - 313. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-201. - 169. i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (31.4 + 86.2i)T + (-3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 - 1.06e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (239. + 285. i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-1.17e3 + 427. i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-781. - 4.43e3i)T + (-1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-4.81e3 + 2.78e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (1.76e3 - 3.05e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.74e3 + 9.87e3i)T + (-3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (4.46e3 + 786. i)T + (4.45e7 + 1.62e7i)T^{2} \) |
| 89 | \( 1 + (1.15e4 + 6.69e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (5.33e3 + 4.47e3i)T + (1.53e7 + 8.71e7i)T^{2} \) |
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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
−12.96213241101219903056754908655, −12.45784294408584929462604663760, −11.62337485590119855780741537526, −9.578317007991173365118941944741, −8.727103955713331332041789194223, −7.14440656753683139051549377736, −5.61364641631142704832598717568, −4.54111308172788664679019744980, −3.38285437732051728927959970205, −0.33874055151679496909328041356,
3.33521721662381555418573806890, 3.55203299543703793674405232406, 5.57611075367816139351813054527, 6.88543293814482097857795676540, 8.066748977232059623134665964173, 9.921395797536457546782156077574, 11.02364837631669613292619269161, 12.04631816550295951084290831219, 12.97971679971073028599644325969, 14.01199430064227496436337272515
