| L(s) = 1 | + (−0.580 + 0.814i)2-s + (−0.235 + 0.971i)3-s + (−0.327 − 0.945i)4-s + (0.170 − 3.58i)5-s + (−0.654 − 0.755i)6-s + (2.64 − 0.0174i)7-s + (0.959 + 0.281i)8-s + (−0.888 − 0.458i)9-s + (2.82 + 2.21i)10-s + (−1.84 − 2.58i)11-s + (0.995 − 0.0950i)12-s + (0.629 − 4.38i)13-s + (−1.52 + 2.16i)14-s + (3.44 + 1.01i)15-s + (−0.786 + 0.618i)16-s + (−6.09 + 1.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.410 + 0.575i)2-s + (−0.136 + 0.561i)3-s + (−0.163 − 0.472i)4-s + (0.0763 − 1.60i)5-s + (−0.267 − 0.308i)6-s + (0.999 − 0.00658i)7-s + (0.339 + 0.0996i)8-s + (−0.296 − 0.152i)9-s + (0.892 + 0.701i)10-s + (−0.555 − 0.779i)11-s + (0.287 − 0.0274i)12-s + (0.174 − 1.21i)13-s + (−0.406 + 0.578i)14-s + (0.889 + 0.261i)15-s + (−0.196 + 0.154i)16-s + (−1.47 + 0.284i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.526853 - 0.591779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.526853 - 0.591779i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.580 - 0.814i)T \) |
| 3 | \( 1 + (0.235 - 0.971i)T \) |
| 7 | \( 1 + (-2.64 + 0.0174i)T \) |
| 23 | \( 1 + (2.91 - 3.80i)T \) |
| good | 5 | \( 1 + (-0.170 + 3.58i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (1.84 + 2.58i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (-0.629 + 4.38i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (6.09 - 1.17i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (-1.35 - 0.260i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (4.44 + 5.13i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (5.80 - 5.53i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (5.12 + 2.64i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (-3.89 - 2.50i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (1.33 - 0.392i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-1.16 - 2.01i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.50 + 1.40i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (2.74 + 2.15i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (2.66 + 10.9i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (-4.57 - 0.436i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (2.78 + 6.10i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-3.16 - 9.13i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (-15.7 + 6.31i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (-11.4 + 7.36i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (0.130 + 0.124i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (-15.7 - 10.1i)T + (40.2 + 88.2i)T^{2} \) |
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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
−9.485516841430668544925418078893, −8.895122962894576098989233647427, −8.177683561430658555260274352991, −7.71064563135084663798071735947, −6.07418259510573450931288090540, −5.31701627157376475591597476585, −4.86969077250482703560730800991, −3.74375386834888043048991743605, −1.81688189127659331213106345340, −0.40970539600350979841613980034,
1.97429780269575050584866407042, 2.34978979285788465782096797011, 3.81288661200448456884817224006, 4.86756877873222023178771198324, 6.24421734839968357962319481472, 7.14144924787215064823558359376, 7.47984343034450490187972785624, 8.634188538858107869026103380249, 9.468204265459240313300156052042, 10.58271086735816283173094695318
