L(s) = 1 | + (12.1 − 6.99i)2-s + (43.2 − 74.9i)3-s + (33.9 − 58.8i)4-s + (−221. − 127. i)5-s − 1.21e3i·6-s + (−32.7 + 56.6i)7-s + 840. i·8-s + (−2.65e3 − 4.59e3i)9-s − 3.57e3·10-s + 4.26e3·11-s + (−2.94e3 − 5.09e3i)12-s + (3.54e3 + 2.04e3i)13-s + 916. i·14-s + (−1.91e4 + 1.10e4i)15-s + (1.02e4 + 1.77e4i)16-s + (8.92e3 − 5.15e3i)17-s + ⋯ |
L(s) = 1 | + (1.07 − 0.618i)2-s + (0.925 − 1.60i)3-s + (0.265 − 0.459i)4-s + (−0.792 − 0.457i)5-s − 2.29i·6-s + (−0.0360 + 0.0624i)7-s + 0.580i·8-s + (−1.21 − 2.10i)9-s − 1.13·10-s + 0.966·11-s + (−0.491 − 0.850i)12-s + (0.447 + 0.258i)13-s + 0.0892i·14-s + (−1.46 + 0.846i)15-s + (0.624 + 1.08i)16-s + (0.440 − 0.254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.07774 - 3.33011i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07774 - 3.33011i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-2.32e5 - 2.02e5i)T \) |
good | 2 | \( 1 + (-12.1 + 6.99i)T + (64 - 110. i)T^{2} \) |
| 3 | \( 1 + (-43.2 + 74.9i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (221. + 127. i)T + (3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (32.7 - 56.6i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 4.26e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-3.54e3 - 2.04e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-8.92e3 + 5.15e3i)T + (2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-5.49e3 - 3.17e3i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + 1.08e5iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 1.79e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 2.30e5iT - 2.75e10T^{2} \) |
| 41 | \( 1 + (2.11e5 - 3.67e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 - 6.92e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 9.64e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (7.39e4 + 1.28e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-4.76e5 + 2.74e5i)T + (1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-6.34e4 - 3.66e4i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.07e6 - 1.86e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-3.53e4 + 6.13e4i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 2.79e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (3.45e6 + 1.99e6i)T + (9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-2.19e6 - 3.79e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.28e6 - 1.89e6i)T + (2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.70e7iT - 8.07e13T^{2} \) |
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
−14.11424468414798961661185825755, −13.02023566605481562649576501012, −12.27916392491467417100441716125, −11.53738121430884711649445461742, −8.840796097429797616385930904049, −7.896425800367482311734793931553, −6.35948777131516562085788756947, −4.11862861949238782645262296576, −2.74328168898941976096190395002, −1.15031774077303998993101635870,
3.47273108121455981739905497407, 3.99494263055333917433716092314, 5.52201426664377029024066087439, 7.48950603691894807955099422935, 9.068729899073905226857927682961, 10.27081270788467586719739968434, 11.68324020814555174401987592956, 13.62916632531510677287880628682, 14.37825598792497408257817562343, 15.44069983325834787286623005824