| L(s) = 1 | + (2 − 3.46i)2-s + (−13.9 + 24.1i)3-s + (−7.99 − 13.8i)4-s + (34.6 − 59.9i)5-s + (55.6 + 96.4i)6-s − 195.·7-s − 63.9·8-s + (−265. − 460. i)9-s + (−138. − 239. i)10-s − 137.·11-s + 445.·12-s + (−398. − 689. i)13-s + (−390. + 676. i)14-s + (963. + 1.66e3i)15-s + (−128 + 221. i)16-s + (−784. + 1.35e3i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.892 + 1.54i)3-s + (−0.249 − 0.433i)4-s + (0.619 − 1.07i)5-s + (0.631 + 1.09i)6-s − 1.50·7-s − 0.353·8-s + (−1.09 − 1.89i)9-s + (−0.437 − 0.758i)10-s − 0.342·11-s + 0.892·12-s + (−0.653 − 1.13i)13-s + (−0.532 + 0.922i)14-s + (1.10 + 1.91i)15-s + (−0.125 + 0.216i)16-s + (−0.658 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0530660 - 0.305182i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0530660 - 0.305182i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 + 3.46i)T \) |
| 19 | \( 1 + (-1.55e3 + 206. i)T \) |
| good | 3 | \( 1 + (13.9 - 24.1i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-34.6 + 59.9i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + 195.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 137.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (398. + 689. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (784. - 1.35e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (-208. - 361. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-155. - 270. i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 1.03e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.32e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-2.54e3 + 4.39e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (4.08e3 - 7.08e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.01e3 + 1.76e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (6.07e3 + 1.05e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.36e3 + 4.09e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (4.31e3 + 7.46e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (6.73e3 + 1.16e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-3.50e4 + 6.07e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-9.04e3 + 1.56e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-4.40e4 + 7.62e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 1.15e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.29e4 + 2.24e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (6.50e4 - 1.12e5i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−15.05745364804765564191304500621, −13.12434781651442484523730952931, −12.44499753745067016799010062707, −10.83242781610871028913943741920, −9.862771216679687195400064177307, −9.204129352906288044298025136738, −5.91360057966664102129150415400, −5.04077866330552632402432359719, −3.46872945919473612392765571808, −0.16426761304176881503812719366,
2.59808515315668009418645366336, 5.65502959465808554366332866337, 6.81280542406333285081417133425, 7.14628014311337779732830965922, 9.573499614307823748974717426921, 11.29185923261483164501972274759, 12.49776511931535850433727901666, 13.44872149077798826194593006468, 14.22637811935392637148645222021, 16.07285377255727331384213183893
