| L(s) = 1 | + (4 − 6.92i)2-s + (39.6 − 68.6i)3-s + (−31.9 − 55.4i)4-s + (99.4 − 172. i)5-s + (−317. − 549. i)6-s + 1.59e3·7-s − 511.·8-s + (−2.05e3 − 3.55e3i)9-s + (−795. − 1.37e3i)10-s − 1.68e3·11-s − 5.07e3·12-s + (7.59e3 + 1.31e4i)13-s + (6.36e3 − 1.10e4i)14-s + (−7.89e3 − 1.36e4i)15-s + (−2.04e3 + 3.54e3i)16-s + (−1.07e4 + 1.86e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.848 − 1.46i)3-s + (−0.249 − 0.433i)4-s + (0.355 − 0.616i)5-s + (−0.599 − 1.03i)6-s + 1.75·7-s − 0.353·8-s + (−0.938 − 1.62i)9-s + (−0.251 − 0.435i)10-s − 0.380·11-s − 0.848·12-s + (0.958 + 1.66i)13-s + (0.619 − 1.07i)14-s + (−0.603 − 1.04i)15-s + (−0.125 + 0.216i)16-s + (−0.531 + 0.920i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.13760 - 2.98028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13760 - 2.98028i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 + 6.92i)T \) |
| 19 | \( 1 + (2.59e4 - 1.47e4i)T \) |
| good | 3 | \( 1 + (-39.6 + 68.6i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-99.4 + 172. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 - 1.59e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.68e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-7.59e3 - 1.31e4i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.07e4 - 1.86e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (3.63e4 + 6.29e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-1.49e4 - 2.58e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 5.36e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.36e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (3.45e5 - 5.97e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-2.20e5 + 3.82e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (1.82e4 + 3.15e4i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (3.43e5 + 5.94e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (4.14e5 - 7.17e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (7.35e5 + 1.27e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.86e6 + 3.22e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.17e6 + 2.03e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (5.58e5 - 9.66e5i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (5.35e5 - 9.26e5i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 6.69e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (1.38e6 + 2.40e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (8.42e5 - 1.45e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−14.04150700492789165754189674578, −13.19742615923465929318267990871, −12.14577711807194245456944580772, −11.01511926187915212911573265170, −8.801055700482688948684624175423, −8.168948749076937456121456601408, −6.35850642852092845152481057442, −4.40906354016818359500965029872, −1.99601157744214582847555339045, −1.44521307340803983067578243129,
2.74680093844361424052457815283, 4.34441190203401847822843602970, 5.49451487022802512902071704115, 7.83985435583529260317010710585, 8.717505098478409603249620368721, 10.32568497726349087653074174319, 11.18877886948464519171072320954, 13.51869865757509403727315493259, 14.35810902863450902845157597500, 15.24408053423684779882599178516
