L(s) = 1 | + (−1.18 + 1.18i)2-s + (5.11 + 0.932i)3-s + 5.17i·4-s + (−2.48 − 10.9i)5-s + (−7.17 + 4.96i)6-s + (−13.3 − 13.3i)7-s + (−15.6 − 15.6i)8-s + (25.2 + 9.53i)9-s + (15.8 + 10i)10-s + 28.7i·11-s + (−4.83 + 26.4i)12-s + (14.1 − 14.1i)13-s + 31.7·14-s + (−2.51 − 58.0i)15-s − 4.25·16-s + (−18.5 + 18.5i)17-s + ⋯ |
L(s) = 1 | + (−0.419 + 0.419i)2-s + (0.983 + 0.179i)3-s + 0.647i·4-s + (−0.221 − 0.975i)5-s + (−0.488 + 0.337i)6-s + (−0.721 − 0.721i)7-s + (−0.691 − 0.691i)8-s + (0.935 + 0.353i)9-s + (0.502 + 0.316i)10-s + 0.787i·11-s + (−0.116 + 0.636i)12-s + (0.302 − 0.302i)13-s + 0.605·14-s + (−0.0433 − 0.999i)15-s − 0.0664·16-s + (−0.264 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.562i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.827 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.933936 + 0.287312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.933936 + 0.287312i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-5.11 - 0.932i)T \) |
| 5 | \( 1 + (2.48 + 10.9i)T \) |
good | 2 | \( 1 + (1.18 - 1.18i)T - 8iT^{2} \) |
| 7 | \( 1 + (13.3 + 13.3i)T + 343iT^{2} \) |
| 11 | \( 1 - 28.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-14.1 + 14.1i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (18.5 - 18.5i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 - 49.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-37.7 - 37.7i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (127. + 127. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 390. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (39.3 - 39.3i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (124. - 124. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-160. - 160. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 729.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 2T + 2.26e5T^{2} \) |
| 67 | \( 1 + (329. + 329. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 171. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (279. - 279. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 48.0iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (144. + 144. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-908. - 908. i)T + 9.12e5iT^{2} \) |
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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
−19.19300339549821438778472856279, −17.47615302026504461901797489048, −16.33759711317536899547976342217, −15.39563343738707901094380443663, −13.43054321387428737362085751487, −12.49743549338424276033983434725, −9.832573702229850873819797035896, −8.576976398107080362336417453962, −7.30324020632429952807597508995, −3.90055542462926303190335709383,
2.82304810281741027661711502846, 6.50255990905518713887888240877, 8.688393218705620137484404251792, 9.964144928199166960579502353493, 11.49630306096740907800884784680, 13.51618850146190279934183483122, 14.77626857759741053425400027435, 15.73209268883751784536857699872, 18.22344697396416525847690746252, 18.98606645131647495282953597326