L(s) = 1 | + (−0.322 − 0.558i)2-s + (2.09 − 3.62i)3-s + (3.79 − 6.56i)4-s + (2.5 + 4.33i)5-s − 2.69·6-s − 10.0·8-s + (4.73 + 8.20i)9-s + (1.61 − 2.79i)10-s + (23.8 − 41.3i)11-s + (−15.8 − 27.5i)12-s + 57.2·13-s + 20.9·15-s + (−27.0 − 46.9i)16-s + (18.4 − 32.0i)17-s + (3.05 − 5.28i)18-s + (−15.3 − 26.6i)19-s + ⋯ |
L(s) = 1 | + (−0.113 − 0.197i)2-s + (0.402 − 0.697i)3-s + (0.474 − 0.821i)4-s + (0.223 + 0.387i)5-s − 0.183·6-s − 0.443·8-s + (0.175 + 0.303i)9-s + (0.0509 − 0.0882i)10-s + (0.653 − 1.13i)11-s + (−0.381 − 0.661i)12-s + 1.22·13-s + 0.360·15-s + (−0.423 − 0.733i)16-s + (0.263 − 0.456i)17-s + (0.0399 − 0.0692i)18-s + (−0.185 − 0.321i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.47625 - 1.80438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47625 - 1.80438i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.5 - 4.33i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.322 + 0.558i)T + (-4 + 6.92i)T^{2} \) |
| 3 | \( 1 + (-2.09 + 3.62i)T + (-13.5 - 23.3i)T^{2} \) |
| 11 | \( 1 + (-23.8 + 41.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 57.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-18.4 + 32.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (15.3 + 26.6i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (26.5 + 46.0i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 195.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (128. - 223. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (173. + 300. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 267.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-155. - 269. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-246. + 426. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (49.3 - 85.5i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (41.0 + 71.1i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (327. - 566. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 779.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-414. + 718. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-384. - 666. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 613.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-228. - 396. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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
−11.01602783797659481197740489977, −10.80477981670143505168051717546, −9.384309004285445323304429815581, −8.529172550331424127880883556670, −7.23685278159238235542919234113, −6.39231069540686680812204655585, −5.45890186336376936992748025650, −3.50639038593507013442600856207, −2.14076073866549596491614964210, −0.995678193554231483450179014712,
1.77813136319889633544888917788, 3.54631831152467129679751485237, 4.20500829324218963608510228327, 5.92242575693958777993638550369, 6.99767074911004328141942932860, 8.116541727616145899438414949052, 9.046434402422408114740372969508, 9.749715278759139720363050697120, 10.94613287648961946105877345301, 12.01720917541955143719361275205