L(s) = 1 | + (7.43 − 12.8i)2-s + (−34.7 − 60.2i)3-s + (−46.6 − 80.7i)4-s + (−238. − 412. i)5-s − 1.03e3·6-s + (249. + 431. i)7-s + 516.·8-s + (−1.32e3 + 2.29e3i)9-s − 7.08e3·10-s + 7.32e3·11-s + (−3.24e3 + 5.62e3i)12-s + (−5.10e3 − 8.83e3i)13-s + 7.41e3·14-s + (−1.65e4 + 2.87e4i)15-s + (9.81e3 − 1.69e4i)16-s + (−1.40e4 + 2.43e4i)17-s + ⋯ |
L(s) = 1 | + (0.657 − 1.13i)2-s + (−0.743 − 1.28i)3-s + (−0.364 − 0.631i)4-s + (−0.852 − 1.47i)5-s − 1.95·6-s + (0.274 + 0.475i)7-s + 0.356·8-s + (−0.607 + 1.05i)9-s − 2.24·10-s + 1.66·11-s + (−0.542 + 0.939i)12-s + (−0.644 − 1.11i)13-s + 0.722·14-s + (−1.26 + 2.19i)15-s + (0.598 − 1.03i)16-s + (−0.692 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.808159 + 1.35266i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.808159 + 1.35266i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + (-1.98e5 + 2.35e5i)T \) |
good | 2 | \( 1 + (-7.43 + 12.8i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (34.7 + 60.2i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (238. + 412. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-249. - 431. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 - 7.32e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (5.10e3 + 8.83e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.40e4 - 2.43e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.08e3 + 3.61e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 - 6.04e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.65e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.91e4T + 2.75e10T^{2} \) |
| 41 | \( 1 + (2.53e5 + 4.38e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 - 1.37e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.98e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (3.44e5 - 5.97e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (3.84e5 - 6.66e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-6.06e5 - 1.05e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.41e6 - 2.45e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.28e6 + 2.22e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 - 1.54e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (2.85e6 + 4.95e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.55e5 + 2.68e5i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.03e6 + 1.79e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 7.63e6T + 8.07e13T^{2} \) |
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
−13.19111678495925475761096091309, −12.55807134640077109823026885753, −11.99891438661384520632183990433, −11.13299533511015988197191470559, −8.861134604874756755839214781970, −7.43644840020992368846220007267, −5.51363954742790741873300410215, −4.08382365934795560890520817458, −1.71485780994007168255614004229, −0.66357531979768961037769658234,
3.83831350393547413894768065077, 4.68373093920933451224451335216, 6.51562947283769647213376664514, 7.22964429312596513717766713307, 9.530393965049458854005551070219, 11.05459195937778741628033243089, 11.53161969007454787946611296968, 14.06697040033058518231337483486, 14.74226429472683275355553724307, 15.45357877417486596206300062289