| L(s) = 1 | + (3.87 + 8.05i)2-s + (−6.20 − 9.10i)3-s + (−9.89 + 12.4i)4-s + (38.7 − 41.7i)5-s + (49.2 − 85.2i)6-s + (32.6 − 18.8i)7-s + (419. + 95.7i)8-s + (222. − 565. i)9-s + (485. + 149. i)10-s + (602. + 755. i)11-s + (174. + 13.0i)12-s + (2.61e3 − 805. i)13-s + (278. + 189. i)14-s + (−619. − 93.4i)15-s + (1.08e3 + 4.73e3i)16-s + (−385. + 357. i)17-s + ⋯ |
| L(s) = 1 | + (0.484 + 1.00i)2-s + (−0.229 − 0.337i)3-s + (−0.154 + 0.193i)4-s + (0.309 − 0.333i)5-s + (0.227 − 0.394i)6-s + (0.0951 − 0.0549i)7-s + (0.819 + 0.186i)8-s + (0.304 − 0.775i)9-s + (0.485 + 0.149i)10-s + (0.452 + 0.567i)11-s + (0.100 + 0.00755i)12-s + (1.18 − 0.366i)13-s + (0.101 + 0.0691i)14-s + (−0.183 − 0.0276i)15-s + (0.264 + 1.15i)16-s + (−0.0784 + 0.0727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.45419 + 0.549971i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.45419 + 0.549971i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.55e4 - 4.49e4i)T \) |
| good | 2 | \( 1 + (-3.87 - 8.05i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (6.20 + 9.10i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (-38.7 + 41.7i)T + (-1.16e3 - 1.55e4i)T^{2} \) |
| 7 | \( 1 + (-32.6 + 18.8i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-602. - 755. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-2.61e3 + 805. i)T + (3.98e6 - 2.71e6i)T^{2} \) |
| 17 | \( 1 + (385. - 357. i)T + (1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (2.96e3 - 1.16e3i)T + (3.44e7 - 3.19e7i)T^{2} \) |
| 23 | \( 1 + (449. - 67.7i)T + (1.41e8 - 4.36e7i)T^{2} \) |
| 29 | \( 1 + (-2.67e4 + 3.92e4i)T + (-2.17e8 - 5.53e8i)T^{2} \) |
| 31 | \( 1 + (-2.71e3 + 3.62e4i)T + (-8.77e8 - 1.32e8i)T^{2} \) |
| 37 | \( 1 + (1.58e4 + 9.17e3i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (3.48e4 - 1.67e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (2.49e4 - 3.13e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (6.34e4 + 1.95e4i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (1.92e3 + 8.41e3i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (1.42e5 - 1.06e4i)T + (5.09e10 - 7.67e9i)T^{2} \) |
| 67 | \( 1 + (-8.59e4 - 2.18e5i)T + (-6.63e10 + 6.15e10i)T^{2} \) |
| 71 | \( 1 + (4.16e4 - 2.76e5i)T + (-1.22e11 - 3.77e10i)T^{2} \) |
| 73 | \( 1 + (-6.87e4 - 2.22e5i)T + (-1.25e11 + 8.52e10i)T^{2} \) |
| 79 | \( 1 + (-2.58e5 - 4.47e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (7.19e5 - 4.90e5i)T + (1.19e11 - 3.04e11i)T^{2} \) |
| 89 | \( 1 + (1.81e5 + 2.65e5i)T + (-1.81e11 + 4.62e11i)T^{2} \) |
| 97 | \( 1 + (-3.61e5 - 4.53e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.96387091404418614699430627681, −13.68521455518901727478268279939, −12.78921985877681950017851731034, −11.34282731741890413741748726619, −9.770889809558887672779268562933, −8.154154311168602599059206804706, −6.71083085291063914203721625403, −5.85817640262984916542340079162, −4.22552989045750511750893868396, −1.34996329012660148998262720205,
1.64921862678595375966463153325, 3.36106401924064828264813871812, 4.82316516942772882283715772627, 6.66840735982312295132923250335, 8.542688153570194670574594704520, 10.35468037232597408403121384045, 10.95069776063117497902015697990, 12.09615862424030720622515529172, 13.41560756566834140396248707761, 14.14570848314117170968292414803
