| L(s) = 1 | + (−1.23 − 1.04i)2-s + (0.358 − 0.130i)3-s + (0.107 + 0.608i)4-s + (0.561 − 3.18i)5-s + (−0.580 − 0.211i)6-s + (−1.5 − 2.59i)7-s + (−1.11 + 1.93i)8-s + (−2.18 + 1.83i)9-s + (−4.01 + 3.36i)10-s + (0.809 − 1.40i)11-s + (0.118 + 0.204i)12-s + (−0.939 − 0.342i)13-s + (−0.842 + 4.78i)14-s + (−0.214 − 1.21i)15-s + (4.56 − 1.66i)16-s + (0.585 + 0.491i)17-s + ⋯ |
| L(s) = 1 | + (−0.876 − 0.735i)2-s + (0.207 − 0.0754i)3-s + (0.0536 + 0.304i)4-s + (0.251 − 1.42i)5-s + (−0.237 − 0.0862i)6-s + (−0.566 − 0.981i)7-s + (−0.395 + 0.684i)8-s + (−0.728 + 0.611i)9-s + (−1.26 + 1.06i)10-s + (0.243 − 0.422i)11-s + (0.0340 + 0.0590i)12-s + (−0.260 − 0.0948i)13-s + (−0.225 + 1.27i)14-s + (−0.0554 − 0.314i)15-s + (1.14 − 0.415i)16-s + (0.141 + 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.105774 + 0.567124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105774 + 0.567124i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (1.23 + 1.04i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.358 + 0.130i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.561 + 3.18i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.809 + 1.40i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.939 + 0.342i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.585 - 0.491i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.934 - 5.30i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.77 + 2.32i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.42 + 7.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.85T + 37T^{2} \) |
| 41 | \( 1 + (2.81 - 1.02i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0253 - 0.143i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.29 + 1.92i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.09 - 6.23i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.249 + 0.209i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.77 + 10.0i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.36 + 4.49i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.29 + 7.35i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.54 + 0.926i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-12.6 + 4.58i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (4.23 + 7.33i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.29 + 2.65i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (10.6 + 8.90i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.80152260176112447409813029697, −9.874320961014032470525218082996, −9.183351413718264553400170391021, −8.440652578989670264753195287509, −7.56815037909768537775786675832, −5.89899032135861973095441074282, −4.98474786649367677796267908216, −3.45864395053766320899765730825, −1.84891675082409024603505887834, −0.49698717471709312770196467538,
2.63530042704481999450258990667, 3.48822452953603331859627292940, 5.59952889162458315627561617245, 6.69918136923951767091771264737, 6.93731308935070108080335229911, 8.407805146036732467861299428299, 9.044863075690111604648973112379, 9.877561923426979546814425920804, 10.71153593317077289289094803000, 12.01166799252400027323229885716
