| L(s) = 1 | + (0.342 − 0.939i)2-s + (1.41 − 0.994i)3-s + (−0.766 − 0.642i)4-s + (0.622 − 3.53i)5-s + (−0.449 − 1.67i)6-s + (2.42 + 1.04i)7-s + (−0.866 + 0.500i)8-s + (1.02 − 2.82i)9-s + (−3.10 − 1.79i)10-s + (−4.15 + 0.732i)11-s + (−1.72 − 0.149i)12-s + (2.25 + 6.20i)13-s + (1.81 − 1.92i)14-s + (−2.62 − 5.62i)15-s + (0.173 + 0.984i)16-s + (−2.59 + 4.49i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (0.818 − 0.574i)3-s + (−0.383 − 0.321i)4-s + (0.278 − 1.57i)5-s + (−0.183 − 0.682i)6-s + (0.918 + 0.395i)7-s + (−0.306 + 0.176i)8-s + (0.340 − 0.940i)9-s + (−0.982 − 0.566i)10-s + (−1.25 + 0.220i)11-s + (−0.498 − 0.0432i)12-s + (0.626 + 1.72i)13-s + (0.485 − 0.514i)14-s + (−0.678 − 1.45i)15-s + (0.0434 + 0.246i)16-s + (−0.630 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09844 - 1.71983i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09844 - 1.71983i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 + 0.939i)T \) |
| 3 | \( 1 + (-1.41 + 0.994i)T \) |
| 7 | \( 1 + (-2.42 - 1.04i)T \) |
| good | 5 | \( 1 + (-0.622 + 3.53i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (4.15 - 0.732i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-2.25 - 6.20i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.59 - 4.49i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.98 + 2.30i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.69 - 3.21i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.58 + 4.35i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.80 + 2.15i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (2.76 - 4.78i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.08 + 1.12i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.856 + 4.85i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.00676 + 0.00567i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 3.95iT - 53T^{2} \) |
| 59 | \( 1 + (-0.405 + 2.29i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (1.23 + 1.46i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.97 + 0.719i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (2.82 + 1.63i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.15 - 4.70i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (10.5 + 3.83i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.26 - 0.459i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-5.76 - 9.98i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.42 + 0.427i)T + (91.1 - 33.1i)T^{2} \) |
| 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
−11.43027859314081932031206830501, −9.986242060372635963570141208577, −9.008850104661212069645206675611, −8.574955425047685742680523581841, −7.67127167742893249700139672422, −6.05111796616826773604758552838, −4.90281814371927735965721347897, −4.07822872943332048082504761249, −2.22948942593653533624004947610, −1.43767230610612150957005108764,
2.65629268835104136105176338016, 3.38121824775630730619524815355, 4.86525931517930217592509365707, 5.78945440902445146040221304718, 7.25351125797375469927592569591, 7.77100477860591555575698981945, 8.603055450776392921039755146688, 10.10873740734012777609621742878, 10.50005594746126759367601807440, 11.29553420429308729468630007287
