| L(s) = 1 | + (2.44 − 1.41i)2-s + (−11.7 + 6.76i)3-s + (3.99 − 6.92i)4-s + (−15.6 − 27.0i)5-s + (−19.1 + 33.1i)6-s − 61.1·7-s − 22.6i·8-s + (51.0 − 88.4i)9-s + (−76.4 − 44.1i)10-s − 130.·11-s + 108. i·12-s + (252. + 146. i)13-s + (−149. + 86.4i)14-s + (365. + 211. i)15-s + (−32.0 − 55.4i)16-s + (−27.4 − 47.6i)17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (−1.30 + 0.751i)3-s + (0.249 − 0.433i)4-s + (−0.624 − 1.08i)5-s + (−0.531 + 0.920i)6-s − 1.24·7-s − 0.353i·8-s + (0.630 − 1.09i)9-s + (−0.764 − 0.441i)10-s − 1.07·11-s + 0.751i·12-s + (1.49 + 0.864i)13-s + (−0.763 + 0.441i)14-s + (1.62 + 0.938i)15-s + (−0.125 − 0.216i)16-s + (−0.0950 − 0.164i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0416253 - 0.330736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0416253 - 0.330736i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.44 + 1.41i)T \) |
| 19 | \( 1 + (276. + 232. i)T \) |
| good | 3 | \( 1 + (11.7 - 6.76i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (15.6 + 27.0i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + 61.1T + 2.40e3T^{2} \) |
| 11 | \( 1 + 130.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-252. - 146. i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (27.4 + 47.6i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 23 | \( 1 + (-96.8 + 167. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (856. + 494. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 - 1.40e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.68e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.38e3 - 800. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (428. + 741. i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-1.10e3 + 1.91e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (2.03e3 + 1.17e3i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-2.05e3 + 1.18e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.48e3 - 4.31e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (1.33e3 + 772. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-6.85e3 + 3.95e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-430. - 744. i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (6.47e3 - 3.74e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 629.T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-2.40e3 - 1.38e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-1.06e4 + 6.16e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−15.61308387650389080323317225495, −13.34112228369704381801026233067, −12.54117767039523202215082009089, −11.39310773946604411666792510301, −10.42277221621272881466534998613, −8.936292577815401499493590351246, −6.45278526930778304669006582626, −5.15462749078060774103854100282, −3.92456230732114399365732346101, −0.20931298714575000964513280735,
3.37187688113680231543887994587, 5.79537476251453493953620316452, 6.57327014829705673363341201532, 7.79805506111566428368508744452, 10.52903209257846962397232860496, 11.31914494104753140064908070364, 12.73377642810421635935392441911, 13.28882105738176978714570832295, 15.19224928565905697156592196789, 15.96769424104179166699358171907
