| 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 | \( 1 + (3.06e4 - 9.43e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−12.03509277426557677876680261391, −11.26741682818090040798913778932, −10.21829569260836419617015017068, −9.621837990521397620509229416053, −8.076995697858990880287979390467, −6.28275233286655780302829560383, −5.85452617715698776915274478948, −4.09264945753902303222423967701, −3.16476041854105764817663246865, −1.48026136720794680961823298648,
0.890665494491683455188702904578, 2.46918184492362410523429854103, 4.38091483609911850642949979709, 5.37766134987832586749772574890, 6.56748882572239104578740973881, 7.45576772700610606596354779735, 8.877067727616214047581611047016, 9.683786821424089975114372050027, 11.39095630663819755767785126901, 12.23396341608579019048190450888
