L(s) = 1 | + (−0.785 + 1.17i)2-s + (1.08 − 1.87i)3-s + (−0.767 − 1.84i)4-s + (−0.5 + 0.866i)5-s + (1.35 + 2.74i)6-s + 3.94i·7-s + (2.77 + 0.548i)8-s + (−0.844 − 1.46i)9-s + (−0.626 − 1.26i)10-s + 3.27i·11-s + (−4.29 − 0.561i)12-s + (2.48 − 1.43i)13-s + (−4.64 − 3.09i)14-s + (1.08 + 1.87i)15-s + (−2.82 + 2.83i)16-s + (2.44 − 4.24i)17-s + ⋯ |
L(s) = 1 | + (−0.555 + 0.831i)2-s + (0.625 − 1.08i)3-s + (−0.383 − 0.923i)4-s + (−0.223 + 0.387i)5-s + (0.553 + 1.12i)6-s + 1.49i·7-s + (0.981 + 0.193i)8-s + (−0.281 − 0.487i)9-s + (−0.197 − 0.401i)10-s + 0.986i·11-s + (−1.23 − 0.162i)12-s + (0.689 − 0.398i)13-s + (−1.24 − 0.828i)14-s + (0.279 + 0.484i)15-s + (−0.705 + 0.708i)16-s + (0.593 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08061 + 0.544871i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08061 + 0.544871i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.785 - 1.17i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-3.58 - 2.47i)T \) |
good | 3 | \( 1 + (-1.08 + 1.87i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 3.94iT - 7T^{2} \) |
| 11 | \( 1 - 3.27iT - 11T^{2} \) |
| 13 | \( 1 + (-2.48 + 1.43i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.44 + 4.24i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4.53 - 2.61i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.15 + 1.82i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 5.70iT - 37T^{2} \) |
| 41 | \( 1 + (5.08 + 2.93i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (8.05 + 4.65i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.70 - 2.13i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.30 - 3.63i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.14 - 3.72i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.29 - 7.44i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.43 + 9.41i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.83 + 4.90i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.15 + 8.92i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.15 + 2.00i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 8.63iT - 83T^{2} \) |
| 89 | \( 1 + (3.11 - 1.80i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.05 + 3.49i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.82210628345960113389670697895, −10.19347883787436686676245289404, −9.465517153165459682012441642007, −8.300102618350016070543386945420, −7.929997614769055598587948057347, −6.92065331529347831669913747489, −6.05827943579215218795723019426, −4.97134826258019483976327023414, −2.90383464048925645354545196134, −1.62569127857075840581769174211,
1.06608930393224093603276272067, 3.24647408364644649328634403899, 3.87065867432408172786189837987, 4.71370200172601678488537376230, 6.64481926928904894184308317179, 8.145536366698065402172297303398, 8.432460077283907434250897399912, 9.667057312413603194916697847194, 10.19999226992397407121886952925, 10.95202363532466489682471051132