| 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
−14.14570848314117170968292414803, −13.41560756566834140396248707761, −12.09615862424030720622515529172, −10.95069776063117497902015697990, −10.35468037232597408403121384045, −8.542688153570194670574594704520, −6.66840735982312295132923250335, −4.82316516942772882283715772627, −3.36106401924064828264813871812, −1.64921862678595375966463153325,
1.34996329012660148998262720205, 4.22552989045750511750893868396, 5.85817640262984916542340079162, 6.71083085291063914203721625403, 8.154154311168602599059206804706, 9.770889809558887672779268562933, 11.34282731741890413741748726619, 12.78921985877681950017851731034, 13.68521455518901727478268279939, 14.96387091404418614699430627681
