| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.909 − 0.415i)3-s + (−0.959 + 0.281i)4-s + (0.880 − 1.01i)5-s + (−0.281 + 0.959i)6-s + (−1.33 + 2.28i)7-s + (0.415 + 0.909i)8-s + (0.654 + 0.755i)9-s + (−1.13 − 0.726i)10-s + (−1.76 − 0.253i)11-s + (0.989 + 0.142i)12-s + (2.23 − 3.47i)13-s + (2.44 + 1.00i)14-s + (−1.22 + 0.558i)15-s + (0.841 − 0.540i)16-s + (−0.880 − 0.258i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.525 − 0.239i)3-s + (−0.479 + 0.140i)4-s + (0.393 − 0.454i)5-s + (−0.115 + 0.391i)6-s + (−0.506 + 0.862i)7-s + (0.146 + 0.321i)8-s + (0.218 + 0.251i)9-s + (−0.357 − 0.229i)10-s + (−0.530 − 0.0763i)11-s + (0.285 + 0.0410i)12-s + (0.619 − 0.963i)13-s + (0.654 + 0.267i)14-s + (−0.315 + 0.144i)15-s + (0.210 − 0.135i)16-s + (−0.213 − 0.0626i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.370161 + 0.270925i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.370161 + 0.270925i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (1.33 - 2.28i)T \) |
| 23 | \( 1 + (2.31 - 4.19i)T \) |
| good | 5 | \( 1 + (-0.880 + 1.01i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (1.76 + 0.253i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.23 + 3.47i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.880 + 0.258i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (7.64 - 2.24i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-3.81 - 1.11i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (4.71 - 2.15i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (8.21 - 7.11i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.77 - 4.14i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.97 - 0.900i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 8.50iT - 47T^{2} \) |
| 53 | \( 1 + (-6.09 - 9.48i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (5.76 - 8.97i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.00 - 2.19i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-8.49 + 1.22i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.145 - 1.01i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (3.04 + 10.3i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-4.26 + 6.63i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-7.16 - 8.27i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (0.718 - 1.57i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (10.4 - 12.1i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.46795957955625548207527682058, −9.385712125221244330262304592044, −8.661912339854011559859595628839, −7.936306993564678041757961257252, −6.58095183090567019035641874331, −5.70904784885583867144071237454, −5.12817796068153696456247531843, −3.78121447145722398103511711731, −2.62867626103319242203795662605, −1.47869953948725207412075013395,
0.23389926239484589480060451630, 2.22496388587191269963671763047, 3.90288308913713421327085495342, 4.50342737517843853452125802570, 5.77312186448937658474195993745, 6.60996278766568709089527506299, 6.90168795204451690843302494811, 8.164921328263241724394060267008, 9.012144316104981409878063428468, 9.909463923270180975151366854175
