| L(s) = 1 | + (2.78 − 0.491i)2-s + (−8.29 − 3.48i)3-s + (7.51 − 2.73i)4-s + (12.0 − 14.3i)5-s + (−24.8 − 5.63i)6-s + (−79.7 − 29.0i)7-s + (19.5 − 11.3i)8-s + (56.7 + 57.8i)9-s + (26.5 − 46.0i)10-s + (−111. − 132. i)11-s + (−71.9 − 3.49i)12-s + (37.4 − 212. i)13-s + (−236. − 41.6i)14-s + (−150. + 77.3i)15-s + (49.0 − 41.1i)16-s + (194. + 112. i)17-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (−0.921 − 0.387i)3-s + (0.469 − 0.171i)4-s + (0.483 − 0.575i)5-s + (−0.689 − 0.156i)6-s + (−1.62 − 0.592i)7-s + (0.306 − 0.176i)8-s + (0.700 + 0.714i)9-s + (0.265 − 0.460i)10-s + (−0.918 − 1.09i)11-s + (−0.499 − 0.0242i)12-s + (0.221 − 1.25i)13-s + (−1.20 − 0.212i)14-s + (−0.668 + 0.343i)15-s + (0.191 − 0.160i)16-s + (0.673 + 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.818i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.573 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.612307 - 1.17680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.612307 - 1.17680i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.78 + 0.491i)T \) |
| 3 | \( 1 + (8.29 + 3.48i)T \) |
| good | 5 | \( 1 + (-12.0 + 14.3i)T + (-108. - 615. i)T^{2} \) |
| 7 | \( 1 + (79.7 + 29.0i)T + (1.83e3 + 1.54e3i)T^{2} \) |
| 11 | \( 1 + (111. + 132. i)T + (-2.54e3 + 1.44e4i)T^{2} \) |
| 13 | \( 1 + (-37.4 + 212. i)T + (-2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (-194. - 112. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-154. - 267. i)T + (-6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-80.4 - 221. i)T + (-2.14e5 + 1.79e5i)T^{2} \) |
| 29 | \( 1 + (290. - 51.2i)T + (6.64e5 - 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-1.61e3 + 586. i)T + (7.07e5 - 5.93e5i)T^{2} \) |
| 37 | \( 1 + (-792. + 1.37e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (1.95e3 + 345. i)T + (2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-816. + 684. i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (107. - 294. i)T + (-3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + 3.68e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (3.32e3 - 3.96e3i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (1.82e3 + 663. i)T + (1.06e7 + 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-944. + 5.35e3i)T + (-1.89e7 - 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-147. - 85.0i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-1.14e3 - 1.97e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (412. + 2.34e3i)T + (-3.66e7 + 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-2.57e3 + 453. i)T + (4.45e7 - 1.62e7i)T^{2} \) |
| 89 | \( 1 + (-8.58e3 + 4.95e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (4.11e3 - 3.44e3i)T + (1.53e7 - 8.71e7i)T^{2} \) |
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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
−13.51745640989651728027022145880, −13.17704355840960468008758470892, −12.27618632874738470176534787587, −10.72151534966346347354444886755, −9.920926777263468512428012006863, −7.74678621388745558581136911702, −6.14018339404213826991259543704, −5.46122544813540015145980181230, −3.33659547503824717576390419269, −0.69384172499918783478854421610,
2.85428112322037170123693930766, 4.75793748467359531816145870236, 6.20709793773813109682637432205, 6.91046545252404971593145289703, 9.537324805096923716064352571531, 10.28830756819039622707709490627, 11.79780445186976181164867363191, 12.64108148291783437075600698506, 13.74364775950933123688338618563, 15.25207638319704331643902272826
