| L(s) = 1 | + (0.654 − 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.182 − 0.400i)5-s + (0.142 − 0.989i)6-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.422 − 0.123i)10-s + (−2.68 − 3.09i)11-s + (−0.654 − 0.755i)12-s + (−3.07 − 0.904i)13-s + (−0.415 + 0.909i)14-s + (−0.370 − 0.237i)15-s + (−0.959 + 0.281i)16-s + (1.07 − 7.45i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.0817 − 0.178i)5-s + (0.0580 − 0.404i)6-s + (−0.362 + 0.106i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.133 − 0.0391i)10-s + (−0.809 − 0.934i)11-s + (−0.189 − 0.218i)12-s + (−0.854 − 0.250i)13-s + (−0.111 + 0.243i)14-s + (−0.0955 − 0.0614i)15-s + (−0.239 + 0.0704i)16-s + (0.259 − 1.80i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.331385 - 1.58469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.331385 - 1.58469i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (4.04 - 2.57i)T \) |
| good | 5 | \( 1 + (0.182 + 0.400i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (2.68 + 3.09i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.07 + 0.904i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-1.07 + 7.45i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.865 - 6.02i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.289 + 2.01i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (5.05 + 3.24i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.23 + 2.71i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.486 - 1.06i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-7.85 + 5.04i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + (5.97 - 1.75i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-8.00 - 2.35i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (8.20 + 5.27i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (6.41 - 7.40i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-4.52 + 5.22i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.18 - 15.2i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (2.95 + 0.868i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.64 + 3.60i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-2.23 + 1.43i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.98 - 4.34i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−9.726785335309919273047863387981, −8.975000189691975425490022171058, −7.87915323579707247217782204067, −7.31165924285782977506808030271, −5.93844306371408668517351437934, −5.34427745394312484357354617778, −4.10550923168721377632017976435, −3.06107536701610210860495802989, −2.31808559888252892838578266515, −0.57413214542787875960054924487,
2.17411883266460161316182591183, 3.20364461017534558926657612284, 4.29602042346448407233428092647, 5.01275635289612183972882303365, 6.11522890946204335950992846654, 7.15338033049307770503807254851, 7.64649689214439908310235157810, 8.679168837062543112198707478960, 9.460240862130953122882457679083, 10.35644949766420468386111241475
