| L(s) = 1 | + (7.82 − 5.68i)2-s + (8.34 + 25.6i)3-s + (−10.6 + 32.7i)4-s + (−100. − 73.3i)5-s + (211. + 153. i)6-s + (−177. + 545. i)7-s + (485. + 1.49e3i)8-s + (−589. + 428. i)9-s − 1.20e3·10-s + (−2.51e3 + 3.63e3i)11-s − 928.·12-s + (−2.55e3 + 1.85e3i)13-s + (1.71e3 + 5.27e3i)14-s + (1.04e3 − 3.20e3i)15-s + (8.73e3 + 6.34e3i)16-s + (1.50e4 + 1.09e4i)17-s + ⋯ |
| L(s) = 1 | + (0.691 − 0.502i)2-s + (0.178 + 0.549i)3-s + (−0.0830 + 0.255i)4-s + (−0.361 − 0.262i)5-s + (0.399 + 0.290i)6-s + (−0.195 + 0.600i)7-s + (0.335 + 1.03i)8-s + (−0.269 + 0.195i)9-s − 0.381·10-s + (−0.568 + 0.822i)11-s − 0.155·12-s + (−0.323 + 0.234i)13-s + (0.166 + 0.513i)14-s + (0.0796 − 0.245i)15-s + (0.533 + 0.387i)16-s + (0.741 + 0.538i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.000544 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.000544 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.35501 + 1.35427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35501 + 1.35427i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.34 - 25.6i)T \) |
| 11 | \( 1 + (2.51e3 - 3.63e3i)T \) |
| good | 2 | \( 1 + (-7.82 + 5.68i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (100. + 73.3i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (177. - 545. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (2.55e3 - 1.85e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-1.50e4 - 1.09e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-756. - 2.32e3i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 5.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (1.54e4 - 4.74e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (3.30e3 - 2.40e3i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.26e5 + 3.90e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (1.47e5 + 4.53e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 3.84e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-8.52e4 - 2.62e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (1.01e6 - 7.39e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (6.96e5 - 2.14e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-2.38e6 - 1.73e6i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 3.06e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-7.19e5 - 5.22e5i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-1.34e6 + 4.12e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (8.52e5 - 6.19e5i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-5.72e6 - 4.15e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 - 5.68e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (8.00e6 - 5.81e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
| 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
−15.30661004208016007824688554721, −14.23970344604795369368779675872, −12.76753471967201177337536770606, −12.08795706886192086696889413703, −10.64266896756806272773223704292, −9.093722283014500834712048758173, −7.73425834573669922071193813510, −5.31809456897784992345087470732, −4.04809165156567461261864081178, −2.51230361294241883560798489649,
0.71525905946717645096404504787, 3.35071503551375568136141529347, 5.21486702994641987772849779097, 6.69231212008972895115068526465, 7.84935523358465634384656396391, 9.790359292049670380976202939452, 11.22844004618170430909340148740, 12.89085410436440741706536360988, 13.70887998230086571282847842309, 14.72065268452235070990150337371
