| L(s) = 1 | + (0.987 + 0.156i)2-s + (0.951 + 0.309i)4-s + (1.79 + 3.52i)5-s + (−0.0699 + 0.0428i)7-s + (0.891 + 0.453i)8-s + (1.22 + 3.76i)10-s + (−1.71 − 0.135i)11-s + (−0.0864 − 0.360i)13-s + (−0.0758 + 0.0314i)14-s + (0.809 + 0.587i)16-s + (2.99 + 3.50i)17-s + (0.777 − 3.23i)19-s + (0.618 + 3.90i)20-s + (−1.67 − 0.402i)22-s + (−1.94 + 1.41i)23-s + ⋯ |
| L(s) = 1 | + (0.698 + 0.110i)2-s + (0.475 + 0.154i)4-s + (0.803 + 1.57i)5-s + (−0.0264 + 0.0162i)7-s + (0.315 + 0.160i)8-s + (0.386 + 1.18i)10-s + (−0.518 − 0.0407i)11-s + (−0.0239 − 0.0998i)13-s + (−0.0202 + 0.00839i)14-s + (0.202 + 0.146i)16-s + (0.725 + 0.849i)17-s + (0.178 − 0.742i)19-s + (0.138 + 0.873i)20-s + (−0.357 − 0.0858i)22-s + (−0.405 + 0.294i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.05540 + 1.52263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.05540 + 1.52263i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.987 - 0.156i)T \) |
| 3 | \( 1 \) |
| 41 | \( 1 + (-2.28 + 5.98i)T \) |
| good | 5 | \( 1 + (-1.79 - 3.52i)T + (-2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (0.0699 - 0.0428i)T + (3.17 - 6.23i)T^{2} \) |
| 11 | \( 1 + (1.71 + 0.135i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (0.0864 + 0.360i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-2.99 - 3.50i)T + (-2.65 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.777 + 3.23i)T + (-16.9 - 8.62i)T^{2} \) |
| 23 | \( 1 + (1.94 - 1.41i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.27 + 3.82i)T + (-4.53 - 28.6i)T^{2} \) |
| 31 | \( 1 + (7.34 - 2.38i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.0157 - 0.0484i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-1.27 + 8.04i)T + (-40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-3.83 + 6.25i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (-6.99 - 5.97i)T + (8.29 + 52.3i)T^{2} \) |
| 59 | \( 1 + (-8.51 - 11.7i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.57 + 0.408i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.940 + 0.0740i)T + (66.1 - 10.4i)T^{2} \) |
| 71 | \( 1 + (0.0496 - 0.630i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (-5.62 + 5.62i)T - 73iT^{2} \) |
| 79 | \( 1 + (5.80 + 2.40i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 11.7iT - 83T^{2} \) |
| 89 | \( 1 + (2.62 + 4.28i)T + (-40.4 + 79.2i)T^{2} \) |
| 97 | \( 1 + (0.790 + 10.0i)T + (-95.8 + 15.1i)T^{2} \) |
| 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
−10.47506374416231683409010247735, −10.12221522107332088927158294454, −8.864502279681936319744048960949, −7.54423002877079222915742777751, −7.01738766710114419209029926560, −5.96325237393309802537470704761, −5.48176433199052721700030456404, −3.93165612203746299927501537704, −2.97137468615590292110982520415, −2.07657406220349148545217473841,
1.13649454658492975982471463317, 2.38273431161549677382784988225, 3.83845808875138068183494381712, 5.00813504116385289080033564633, 5.38871935182073235930798398644, 6.38193326397670858991026626089, 7.67605650553929091128085778326, 8.486172433792558708368752522549, 9.548625835748218710384020210071, 9.998749758365633965251948006902
