| L(s) = 1 | + (3.23 + 2.35i)2-s + (−2.78 − 8.55i)3-s + (4.94 + 15.2i)4-s + (32.6 − 45.3i)5-s + (11.1 − 34.2i)6-s − 9.31·7-s + (−19.7 + 60.8i)8-s + (−65.5 + 47.6i)9-s + (212. − 69.9i)10-s + (429. + 312. i)11-s + (116. − 84.6i)12-s + (−16.2 + 11.7i)13-s + (−30.1 − 21.9i)14-s + (−479. − 153. i)15-s + (−207. + 150. i)16-s + (504. − 1.55e3i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (0.584 − 0.811i)5-s + (0.126 − 0.388i)6-s − 0.0718·7-s + (−0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.671 − 0.221i)10-s + (1.07 + 0.778i)11-s + (0.233 − 0.169i)12-s + (−0.0266 + 0.0193i)13-s + (−0.0411 − 0.0298i)14-s + (−0.549 − 0.176i)15-s + (−0.202 + 0.146i)16-s + (0.423 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.827861760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.827861760\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 3 | \( 1 + (2.78 + 8.55i)T \) |
| 5 | \( 1 + (-32.6 + 45.3i)T \) |
| good | 7 | \( 1 + 9.31T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-429. - 312. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (16.2 - 11.7i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-504. + 1.55e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-734. + 2.26e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (1.79e3 + 1.30e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-2.00e3 - 6.18e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.31e3 + 4.03e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-9.25e3 + 6.72e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.31e4 + 9.55e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-656. - 2.02e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (2.38e3 + 7.32e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-5.15e3 + 3.74e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-3.56e4 - 2.58e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.38e4 - 4.26e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (1.58e4 + 4.89e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (4.28e4 + 3.11e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-3.15e4 - 9.72e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.72e4 - 8.40e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (1.92e4 + 1.40e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (3.06e4 + 9.43e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
| 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
−12.23396341608579019048190450888, −11.39095630663819755767785126901, −9.683786821424089975114372050027, −8.877067727616214047581611047016, −7.45576772700610606596354779735, −6.56748882572239104578740973881, −5.37766134987832586749772574890, −4.38091483609911850642949979709, −2.46918184492362410523429854103, −0.890665494491683455188702904578,
1.48026136720794680961823298648, 3.16476041854105764817663246865, 4.09264945753902303222423967701, 5.85452617715698776915274478948, 6.28275233286655780302829560383, 8.076995697858990880287979390467, 9.621837990521397620509229416053, 10.21829569260836419617015017068, 11.26741682818090040798913778932, 12.03509277426557677876680261391
