| L(s) = 1 | + (2.94 + 0.582i)3-s + (5.13 − 3.82i)5-s + (3.31 − 2.17i)7-s + (8.32 + 3.42i)9-s + (−0.480 + 4.11i)11-s + (−0.145 − 0.486i)13-s + (17.3 − 8.25i)15-s + (−19.6 − 3.46i)17-s + (−2.04 − 11.6i)19-s + (11.0 − 4.48i)21-s + (8.81 − 13.4i)23-s + (4.58 − 15.3i)25-s + (22.4 + 14.9i)27-s + (−10.8 + 10.2i)29-s + (−1.57 + 27.1i)31-s + ⋯ |
| L(s) = 1 | + (0.980 + 0.194i)3-s + (1.02 − 0.764i)5-s + (0.473 − 0.311i)7-s + (0.924 + 0.381i)9-s + (−0.0436 + 0.373i)11-s + (−0.0112 − 0.0374i)13-s + (1.15 − 0.550i)15-s + (−1.15 − 0.203i)17-s + (−0.107 − 0.611i)19-s + (0.524 − 0.213i)21-s + (0.383 − 0.583i)23-s + (0.183 − 0.612i)25-s + (0.832 + 0.553i)27-s + (−0.375 + 0.353i)29-s + (−0.0509 + 0.874i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.75291 - 0.411998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.75291 - 0.411998i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.94 - 0.582i)T \) |
| good | 5 | \( 1 + (-5.13 + 3.82i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (-3.31 + 2.17i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (0.480 - 4.11i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (0.145 + 0.486i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (19.6 + 3.46i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (2.04 + 11.6i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (-8.81 + 13.4i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (10.8 - 10.2i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (1.57 - 27.1i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (-11.6 + 4.23i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (-1.01 + 4.29i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (-21.3 - 49.5i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (-25.7 + 1.50i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (5.12 + 2.96i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (8.92 + 76.3i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (13.5 + 6.78i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (68.4 - 72.5i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-5.41 + 6.45i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (24.0 - 20.1i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-50.9 + 12.0i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (14.7 + 62.2i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (108. + 128. i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (42.4 - 57.0i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−11.15492123607258704079842484807, −10.25468637405457350930592662069, −9.235227218514518703719269957771, −8.827380334947461273803978927091, −7.67062434099626761161024069772, −6.59123624060708501835774817906, −5.08484785787951027474856873995, −4.34334775865407278044216711432, −2.64306530897322143751180482424, −1.49032686517571304032858782451,
1.81791153143739790829667049981, 2.71220624564636312750395347287, 4.09231912854468073284336129305, 5.67116030006210665426397597651, 6.63371245855149753431684619176, 7.67219389576980231041195205246, 8.703855391487138424702576427666, 9.486248159165353815533178205583, 10.39789489373713228918900527655, 11.29834051940636829107652378459
