| L(s) = 1 | + (−0.894 − 1.09i)2-s + (3.00 + 0.637i)3-s + (−0.401 + 1.95i)4-s + (−1.91 − 3.30i)5-s + (−1.98 − 3.85i)6-s + (−0.138 − 0.310i)7-s + (2.50 − 1.31i)8-s + (5.85 + 2.60i)9-s + (−1.91 + 5.05i)10-s + (−0.0721 + 0.686i)11-s + (−2.45 + 5.62i)12-s + (2.99 + 2.69i)13-s + (−0.216 + 0.428i)14-s + (−3.62 − 11.1i)15-s + (−3.67 − 1.57i)16-s + (−3.78 + 0.398i)17-s + ⋯ |
| L(s) = 1 | + (−0.632 − 0.774i)2-s + (1.73 + 0.368i)3-s + (−0.200 + 0.979i)4-s + (−0.854 − 1.47i)5-s + (−0.809 − 1.57i)6-s + (−0.0522 − 0.117i)7-s + (0.885 − 0.463i)8-s + (1.95 + 0.869i)9-s + (−0.606 + 1.59i)10-s + (−0.0217 + 0.207i)11-s + (−0.708 + 1.62i)12-s + (0.829 + 0.746i)13-s + (−0.0578 + 0.114i)14-s + (−0.935 − 2.87i)15-s + (−0.919 − 0.393i)16-s + (−0.918 + 0.0965i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02143 - 0.524073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02143 - 0.524073i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.894 + 1.09i)T \) |
| 31 | \( 1 + (4.61 + 3.10i)T \) |
| good | 3 | \( 1 + (-3.00 - 0.637i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (1.91 + 3.30i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.138 + 0.310i)T + (-4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (0.0721 - 0.686i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-2.99 - 2.69i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (3.78 - 0.398i)T + (16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (2.35 - 2.12i)T + (1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (3.13 - 2.28i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 0.363i)T + (23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (2.02 + 1.17i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.99 + 0.848i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-0.595 - 0.660i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-2.48 + 0.807i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.07 - 6.91i)T + (-35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-1.17 + 5.51i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 - 4.62iT - 61T^{2} \) |
| 67 | \( 1 + (-12.3 + 7.13i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.16 + 2.61i)T + (-47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (3.58 + 0.376i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (0.127 + 1.21i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (10.9 - 2.31i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (1.21 - 1.67i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (4.16 + 3.02i)T + (29.9 + 92.2i)T^{2} \) |
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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
−13.15304192427017326500903941038, −12.38420179086961738164754793078, −11.11622158265546604369278308210, −9.708987953246674464710864086123, −8.868345396280116685848658544175, −8.427647535491810170766461905170, −7.46650485779475102590836851677, −4.34382980632779906900362783575, −3.78255394025905706700776517266, −1.86914967622685656474565147108,
2.53869118748610476794760964789, 3.89699767454252661410396064166, 6.44768591322031036534952560031, 7.29438949545080059870766349542, 8.172601203143001214616460410221, 8.895379895415677360941856920923, 10.25330115644814156154911577898, 11.12829999158168765447961337310, 12.99382049191742046972773530705, 14.04590700171114268096282764165
