| L(s) = 1 | + (1.40 + 0.169i)2-s + (0.945 − 2.59i)3-s + (1.94 + 0.475i)4-s + (−1.15 + 0.202i)5-s + (1.76 − 3.48i)6-s + (−2.48 + 4.29i)7-s + (2.64 + 0.995i)8-s + (−3.56 − 2.98i)9-s + (−1.64 + 0.0901i)10-s + (−1.12 + 0.651i)11-s + (3.07 − 4.59i)12-s + (−1.56 − 4.30i)13-s + (−4.21 + 5.61i)14-s + (−0.561 + 3.18i)15-s + (3.54 + 1.84i)16-s + (2.55 − 2.14i)17-s + ⋯ |
| L(s) = 1 | + (0.992 + 0.119i)2-s + (0.546 − 1.50i)3-s + (0.971 + 0.237i)4-s + (−0.514 + 0.0907i)5-s + (0.721 − 1.42i)6-s + (−0.937 + 1.62i)7-s + (0.935 + 0.352i)8-s + (−1.18 − 0.996i)9-s + (−0.521 + 0.0285i)10-s + (−0.340 + 0.196i)11-s + (0.887 − 1.32i)12-s + (−0.434 − 1.19i)13-s + (−1.12 + 1.50i)14-s + (−0.144 + 0.821i)15-s + (0.887 + 0.461i)16-s + (0.619 − 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.84383 - 0.605360i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84383 - 0.605360i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.169i)T \) |
| 19 | \( 1 + (0.369 - 4.34i)T \) |
| good | 3 | \( 1 + (-0.945 + 2.59i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (1.15 - 0.202i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (2.48 - 4.29i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.12 - 0.651i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.56 + 4.30i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.55 + 2.14i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.0598 - 0.339i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.903 + 1.07i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.59 + 4.49i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.14iT - 37T^{2} \) |
| 41 | \( 1 + (-8.84 - 3.22i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (6.63 - 1.17i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.689 + 0.578i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (1.26 + 0.223i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (6.06 + 7.23i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (2.91 + 0.513i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.95 - 2.33i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.96 - 11.1i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.39 + 0.507i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-8.75 - 3.18i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-11.1 - 6.46i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.44 + 0.889i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (2.82 - 2.36i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.62450623915375350883568849176, −12.46404097578479480598850754032, −11.57057704310773876727953156439, −9.783440848860348306270989726562, −8.163931057370069226818172356202, −7.60597812980266496406863794538, −6.31617060963023717255631318863, −5.51847931180106824883186959754, −3.23476958945833844094731760580, −2.36208323879986433244457182115,
3.15779308661372941585163038985, 4.04127740873101347574735761902, 4.74250746595888331639300871751, 6.55398009676421110215793250185, 7.70179834664553794113914486599, 9.338453420997418775704591743685, 10.29247274692883430797038950977, 10.86009022263908548556063326026, 12.11457779828599766885612097407, 13.44135958495603063284539770973
