| 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
−14.04590700171114268096282764165, −12.99382049191742046972773530705, −11.12829999158168765447961337310, −10.25330115644814156154911577898, −8.895379895415677360941856920923, −8.172601203143001214616460410221, −7.29438949545080059870766349542, −6.44768591322031036534952560031, −3.89699767454252661410396064166, −2.53869118748610476794760964789,
1.86914967622685656474565147108, 3.78255394025905706700776517266, 4.34382980632779906900362783575, 7.46650485779475102590836851677, 8.427647535491810170766461905170, 8.868345396280116685848658544175, 9.708987953246674464710864086123, 11.11622158265546604369278308210, 12.38420179086961738164754793078, 13.15304192427017326500903941038
