| L(s) = 1 | + (0.118 + 0.363i)2-s + (−1.53 − 1.70i)3-s + (1.5 − 1.08i)4-s + (1.11 + 1.93i)5-s + (0.437 − 0.756i)6-s + (0.104 + 0.994i)7-s + (1.19 + 0.865i)8-s + (−0.233 + 2.22i)9-s + (−0.571 + 0.634i)10-s + (−3.87 − 1.72i)11-s + (−4.14 − 0.882i)12-s + (6.71 − 1.42i)13-s + (−0.348 + 0.155i)14-s + (1.58 − 4.86i)15-s + (0.972 − 2.99i)16-s + (−0.493 + 0.219i)17-s + ⋯ |
| L(s) = 1 | + (0.0834 + 0.256i)2-s + (−0.884 − 0.981i)3-s + (0.750 − 0.544i)4-s + (0.499 + 0.866i)5-s + (0.178 − 0.309i)6-s + (0.0395 + 0.375i)7-s + (0.421 + 0.305i)8-s + (−0.0779 + 0.741i)9-s + (−0.180 + 0.200i)10-s + (−1.16 − 0.520i)11-s + (−1.19 − 0.254i)12-s + (1.86 − 0.395i)13-s + (−0.0932 + 0.0415i)14-s + (0.408 − 1.25i)15-s + (0.243 − 0.747i)16-s + (−0.119 + 0.0532i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58265 - 0.601517i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58265 - 0.601517i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (-0.118 - 0.363i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (1.53 + 1.70i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (-1.11 - 1.93i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.104 - 0.994i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (3.87 + 1.72i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-6.71 + 1.42i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (0.493 - 0.219i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (0.978 + 0.207i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-5.55 - 4.03i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.14 - 3.52i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-2.12 + 3.67i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.99 + 5.55i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (0.201 + 0.0428i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (-1.14 + 3.52i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.548 + 5.21i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (3.97 + 4.41i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + 4.44T + 61T^{2} \) |
| 67 | \( 1 + (3 + 5.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.781 - 7.43i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (3.87 + 1.72i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-10.1 + 4.51i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (2.11 - 2.35i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (4.74 - 3.44i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.66 + 4.11i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.41737376284565896140019932885, −9.024479446865110122286985504685, −7.939497560712693608490040678403, −7.09734645174707083942588994778, −6.41648973184200407344964824474, −5.79740314150117296653440050621, −5.33086454810069663356209300301, −3.27501168238422078894664196127, −2.21392014085475032176319900478, −1.01569173585309782097498974623,
1.27512625659673273353441279601, 2.76775807594190354117800059656, 4.11997568053009372603734614496, 4.69652202313928187405906796031, 5.74239303171277107269687431777, 6.46475435915178455031853293087, 7.64949490943116680788871161317, 8.557556912611323588345515720442, 9.440680355157894365853913327481, 10.52913120259742212719107127505
