| L(s) = 1 | + (−0.156 − 0.987i)2-s + (3.04 − 0.239i)3-s + (−0.951 + 0.309i)4-s + (0.463 − 0.755i)5-s + (−0.713 − 2.97i)6-s + (0.353 − 4.49i)7-s + (0.453 + 0.891i)8-s + (6.26 − 0.991i)9-s + (−0.819 − 0.339i)10-s + (−1.67 + 2.86i)11-s + (−2.82 + 1.16i)12-s + (−2.70 + 3.71i)13-s + (−4.49 + 0.353i)14-s + (1.23 − 2.41i)15-s + (0.809 − 0.587i)16-s + (−3.00 + 2.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 − 0.698i)2-s + (1.75 − 0.138i)3-s + (−0.475 + 0.154i)4-s + (0.207 − 0.338i)5-s + (−0.291 − 1.21i)6-s + (0.133 − 1.69i)7-s + (0.160 + 0.315i)8-s + (2.08 − 0.330i)9-s + (−0.259 − 0.107i)10-s + (−0.505 + 0.862i)11-s + (−0.815 + 0.337i)12-s + (−0.749 + 1.03i)13-s + (−1.20 + 0.0944i)14-s + (0.317 − 0.623i)15-s + (0.202 − 0.146i)16-s + (−0.728 + 0.685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.62097 - 1.36251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62097 - 1.36251i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.156 + 0.987i)T \) |
| 11 | \( 1 + (1.67 - 2.86i)T \) |
| 17 | \( 1 + (3.00 - 2.82i)T \) |
| good | 3 | \( 1 + (-3.04 + 0.239i)T + (2.96 - 0.469i)T^{2} \) |
| 5 | \( 1 + (-0.463 + 0.755i)T + (-2.26 - 4.45i)T^{2} \) |
| 7 | \( 1 + (-0.353 + 4.49i)T + (-6.91 - 1.09i)T^{2} \) |
| 13 | \( 1 + (2.70 - 3.71i)T + (-4.01 - 12.3i)T^{2} \) |
| 19 | \( 1 + (1.89 - 0.966i)T + (11.1 - 15.3i)T^{2} \) |
| 23 | \( 1 + (-1.56 + 3.77i)T + (-16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-2.60 - 3.05i)T + (-4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-0.632 + 2.63i)T + (-27.6 - 14.0i)T^{2} \) |
| 37 | \( 1 + (-4.93 + 4.21i)T + (5.78 - 36.5i)T^{2} \) |
| 41 | \( 1 + (-2.39 + 2.80i)T + (-6.41 - 40.4i)T^{2} \) |
| 43 | \( 1 + (0.216 - 0.216i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.882 + 0.286i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.71 - 10.8i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-9.19 - 4.68i)T + (34.6 + 47.7i)T^{2} \) |
| 61 | \( 1 + (-8.10 + 1.94i)T + (54.3 - 27.6i)T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + (6.10 + 3.74i)T + (32.2 + 63.2i)T^{2} \) |
| 73 | \( 1 + (8.27 + 9.69i)T + (-11.4 + 72.1i)T^{2} \) |
| 79 | \( 1 + (10.3 - 6.33i)T + (35.8 - 70.3i)T^{2} \) |
| 83 | \( 1 + (0.559 + 0.0886i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 - 12.3iT - 89T^{2} \) |
| 97 | \( 1 + (8.47 + 2.03i)T + (86.4 + 44.0i)T^{2} \) |
| 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
−10.84392373686169144580955198506, −10.14926425336052539709516990162, −9.350552749229348478506996044155, −8.575674190516218979464240352430, −7.53820077316114746089649728604, −6.97631998078987978254447584868, −4.47485067516448647107462423256, −4.10640728241687795649083308999, −2.61977371157779767960546185500, −1.56437644132231441903154049289,
2.47738938491653837073923342218, 3.00145320399046617893305155797, 4.75354907949943939609676137232, 5.82635267561394130290353404603, 7.09723358390522052749803663832, 8.272496604530298759397856579094, 8.507111419542351196295867374519, 9.448420773096516017025885074233, 10.19679499534379159280377900252, 11.62835316415562540112416767550
