| L(s) = 1 | + (1.17 + 0.780i)2-s + (−0.821 + 0.0615i)3-s + (0.780 + 1.84i)4-s + (0.771 + 1.13i)5-s + (−1.01 − 0.568i)6-s + (−2.64 + 0.0535i)7-s + (−0.518 + 2.78i)8-s + (−2.29 + 0.346i)9-s + (0.0259 + 1.93i)10-s + (−0.382 + 2.53i)11-s + (−0.754 − 1.46i)12-s + (2.54 + 2.03i)13-s + (−3.16 − 2.00i)14-s + (−0.703 − 0.882i)15-s + (−2.78 + 2.87i)16-s + (3.13 − 0.967i)17-s + ⋯ |
| L(s) = 1 | + (0.833 + 0.552i)2-s + (−0.474 + 0.0355i)3-s + (0.390 + 0.920i)4-s + (0.345 + 0.506i)5-s + (−0.414 − 0.232i)6-s + (−0.999 + 0.0202i)7-s + (−0.183 + 0.983i)8-s + (−0.765 + 0.115i)9-s + (0.00820 + 0.612i)10-s + (−0.115 + 0.765i)11-s + (−0.217 − 0.422i)12-s + (0.706 + 0.563i)13-s + (−0.844 − 0.535i)14-s + (−0.181 − 0.227i)15-s + (−0.695 + 0.718i)16-s + (0.760 − 0.234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.727438 + 1.42287i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.727438 + 1.42287i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.17 - 0.780i)T \) |
| 7 | \( 1 + (2.64 - 0.0535i)T \) |
| good | 3 | \( 1 + (0.821 - 0.0615i)T + (2.96 - 0.447i)T^{2} \) |
| 5 | \( 1 + (-0.771 - 1.13i)T + (-1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.382 - 2.53i)T + (-10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (-2.54 - 2.03i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-3.13 + 0.967i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.31 - 1.33i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.12 + 1.27i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-4.73 - 1.08i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (0.128 + 0.222i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.36 - 2.54i)T + (-2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-3.87 - 1.86i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (2.53 + 5.27i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-2.21 - 5.64i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-3.66 - 3.95i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (-3.65 + 5.36i)T + (-21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-1.22 + 1.31i)T + (-4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.0323 + 0.0186i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.286 + 1.25i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-4.18 + 10.6i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-3.04 + 5.27i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.8 + 10.2i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (9.74 - 1.46i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 - 8.09T + 97T^{2} \) |
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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
−11.99248775594295409964663783587, −10.81130173450020212828578884381, −9.946236764827170762841075208934, −8.758057434870540402377991025981, −7.61134415302745464145749772456, −6.50228500783289436931418565169, −6.07648795467089093280609171376, −4.96748070272328366408466006624, −3.63382925287275266233485361102, −2.55345343515760739021287009502,
0.870556276497723778706064191941, 2.86453227875108635049948496695, 3.76371924724990691657659371013, 5.45141504848763396196799497733, 5.74701955920918604636158117011, 6.76063064846509005875568570343, 8.360907609382986491555415343177, 9.417748009930500563541465493087, 10.28888213680459952914809809155, 11.12854294990990055275018487438
