| L(s) = 1 | + (−0.315 − 0.229i)2-s + (5.06 − 1.16i)3-s + (−2.42 − 7.46i)4-s + (2.50 + 3.44i)5-s + (−1.86 − 0.793i)6-s + (11.0 − 3.60i)7-s + (−1.90 + 5.87i)8-s + (24.3 − 11.7i)9-s − 1.65i·10-s + (−25.5 + 26.0i)11-s + (−20.9 − 34.9i)12-s + (−19.1 + 26.3i)13-s + (−4.32 − 1.40i)14-s + (16.6 + 14.5i)15-s + (−48.8 + 35.4i)16-s + (−76.8 + 55.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.111 − 0.0809i)2-s + (0.974 − 0.223i)3-s + (−0.303 − 0.933i)4-s + (0.223 + 0.308i)5-s + (−0.126 − 0.0540i)6-s + (0.598 − 0.194i)7-s + (−0.0843 + 0.259i)8-s + (0.900 − 0.435i)9-s − 0.0524i·10-s + (−0.699 + 0.714i)11-s + (−0.504 − 0.841i)12-s + (−0.408 + 0.562i)13-s + (−0.0825 − 0.0268i)14-s + (0.286 + 0.250i)15-s + (−0.763 + 0.554i)16-s + (−1.09 + 0.796i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.41337 - 0.451592i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41337 - 0.451592i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.06 + 1.16i)T \) |
| 11 | \( 1 + (25.5 - 26.0i)T \) |
| good | 2 | \( 1 + (0.315 + 0.229i)T + (2.47 + 7.60i)T^{2} \) |
| 5 | \( 1 + (-2.50 - 3.44i)T + (-38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (-11.0 + 3.60i)T + (277. - 201. i)T^{2} \) |
| 13 | \( 1 + (19.1 - 26.3i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (76.8 - 55.8i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-55.1 - 17.9i)T + (5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 78.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-49.8 - 153. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-35.3 - 25.7i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (27.2 + 83.8i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-146. + 451. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 285. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (525. + 170. i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-165. + 227. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-70.5 + 22.9i)T + (1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (480. + 661. i)T + (-7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 376.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-133. - 184. i)T + (-1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-478. + 155. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (414. - 570. i)T + (-1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (850. - 617. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 661. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-982. - 714. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−15.73275836686394668422063200348, −14.63089393885083858749190137074, −14.05241670133947963786606604355, −12.70988306103211705383949235445, −10.77628190848954411747258872699, −9.722116106224701525532711892302, −8.406582020755878139606087076436, −6.79114304625305071588872490417, −4.66099914170886976238129292441, −2.03220372257332953835605073213,
2.91885249358643194544852936824, 4.82688681587027144728906493588, 7.53716532913133387314981284057, 8.492917022285787051926209714212, 9.604583618417294391969200996871, 11.45052836571177573439661464639, 13.09133430051489867015937476343, 13.70506108841645505766645500065, 15.25927275426281304449050705981, 16.23111807527434742655127143034
