| L(s) = 1 | + (0.104 − 0.994i)2-s + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (0.181 + 0.0808i)5-s + (0.809 − 0.587i)6-s + (0.719 − 2.54i)7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)9-s + (0.0993 − 0.172i)10-s + (3.31 + 0.0523i)11-s + (−0.5 − 0.866i)12-s + (2.44 + 1.77i)13-s + (−2.45 − 0.981i)14-s + (0.0614 + 0.189i)15-s + (0.913 + 0.406i)16-s + (−0.553 − 5.26i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 − 0.703i)2-s + (0.386 + 0.429i)3-s + (−0.489 − 0.103i)4-s + (0.0812 + 0.0361i)5-s + (0.330 − 0.239i)6-s + (0.271 − 0.962i)7-s + (−0.109 + 0.336i)8-s + (−0.0348 + 0.331i)9-s + (0.0314 − 0.0544i)10-s + (0.999 + 0.0157i)11-s + (−0.144 − 0.249i)12-s + (0.676 + 0.491i)13-s + (−0.656 − 0.262i)14-s + (0.0158 + 0.0488i)15-s + (0.228 + 0.101i)16-s + (−0.134 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52466 - 0.792824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52466 - 0.792824i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 + 0.994i)T \) |
| 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 7 | \( 1 + (-0.719 + 2.54i)T \) |
| 11 | \( 1 + (-3.31 - 0.0523i)T \) |
| good | 5 | \( 1 + (-0.181 - 0.0808i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (-2.44 - 1.77i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.553 + 5.26i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.18 + 0.252i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (0.917 + 1.58i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.77 - 8.54i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.66 + 2.07i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-7.75 + 8.60i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (2.45 - 7.54i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + (12.2 - 2.59i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (6.22 - 2.77i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-1.98 - 0.422i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (8.19 + 3.64i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (7.93 - 13.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.72 - 2.70i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (5.56 + 1.18i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (0.851 - 8.09i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (11.6 - 8.42i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.05 - 5.28i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.40 - 1.74i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.05954590983368478317196538446, −9.961400713714538064800449603655, −9.362711749402013488465339339459, −8.430794782120928603454752387091, −7.32878596270999317139385830646, −6.22005281852349158796786738189, −4.66811794089607353628760326683, −4.06000415594997387357204678969, −2.86481399941574673420920100912, −1.24760891928638419273556794083,
1.61825658706622048322278071067, 3.26306640424731584983176445946, 4.48051067275969663580999478843, 5.97934905931452070108610295473, 6.25212306193687952670479146399, 7.70938807790925437449154510903, 8.358073199128162266458635195256, 9.108864617936233952018381623380, 10.02923238842877230496530048094, 11.43182871789902125842742101477
