| L(s) = 1 | + (−1.31 + 0.527i)2-s + (−0.108 + 0.129i)3-s + (1.44 − 1.38i)4-s + (1.67 − 1.48i)5-s + (0.0741 − 0.227i)6-s + (0.176 + 0.305i)7-s + (−1.16 + 2.57i)8-s + (0.515 + 2.92i)9-s + (−1.40 + 2.83i)10-s + (−0.465 − 0.268i)11-s + (0.0226 + 0.337i)12-s + (2.90 − 2.44i)13-s + (−0.392 − 0.307i)14-s + (0.0107 + 0.377i)15-s + (0.162 − 3.99i)16-s + (1.16 + 0.204i)17-s + ⋯ |
| L(s) = 1 | + (−0.927 + 0.373i)2-s + (−0.0626 + 0.0747i)3-s + (0.721 − 0.692i)4-s + (0.747 − 0.664i)5-s + (0.0302 − 0.0927i)6-s + (0.0666 + 0.115i)7-s + (−0.410 + 0.911i)8-s + (0.171 + 0.975i)9-s + (−0.445 + 0.895i)10-s + (−0.140 − 0.0810i)11-s + (0.00652 + 0.0973i)12-s + (0.807 − 0.677i)13-s + (−0.104 − 0.0821i)14-s + (0.00278 + 0.0974i)15-s + (0.0405 − 0.999i)16-s + (0.281 + 0.0497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05269 + 0.0485778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05269 + 0.0485778i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 - 0.527i)T \) |
| 5 | \( 1 + (-1.67 + 1.48i)T \) |
| 19 | \( 1 + (1.57 + 4.06i)T \) |
| good | 3 | \( 1 + (0.108 - 0.129i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-0.176 - 0.305i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.465 + 0.268i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.90 + 2.44i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.16 - 0.204i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-5.38 - 1.96i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-2.19 + 0.386i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.30 - 5.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.773T + 37T^{2} \) |
| 41 | \( 1 + (2.78 - 3.31i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-10.6 + 3.87i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.460 - 2.61i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (4.14 + 1.50i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.583 - 3.30i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (6.49 + 2.36i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (6.98 - 1.23i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (6.72 - 2.44i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.32 + 8.73i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-5.66 - 4.74i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.764 + 1.32i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.25 - 3.87i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.09 - 11.8i)T + (-91.1 - 33.1i)T^{2} \) |
| 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
−10.87015575993350001881526430207, −10.50925886798552258147258539861, −9.381100405453459260503605681731, −8.648363906115510558625601601016, −7.85858063620696206918905877660, −6.68891455718367317166590338137, −5.60804887437279300046207750402, −4.87493450907751563785177109801, −2.66486096314892936097679238660, −1.22932809510414894285931475814,
1.35806138466436173041937184358, 2.79923639280983123155569193114, 3.99427948129228935546634108004, 6.00996958777214092732419269010, 6.63688857409217748504823333187, 7.63359016082312819334317496587, 8.859400286443664361423717086557, 9.512042128315096571715520517104, 10.41658816286783846511028831890, 11.08952133578668462156122052434
