| L(s) = 1 | + (−0.104 + 0.994i)2-s + (−0.0126 + 1.73i)3-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)5-s + (−1.72 − 0.193i)6-s + (2.08 − 2.31i)7-s + (0.309 − 0.951i)8-s + (−2.99 − 0.0437i)9-s + 0.999·10-s + (0.912 + 3.18i)11-s + (0.372 − 1.69i)12-s + (−4.07 + 1.81i)13-s + (2.08 + 2.31i)14-s + (1.72 − 0.168i)15-s + (0.913 + 0.406i)16-s + (−3.50 + 2.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (−0.00729 + 0.999i)3-s + (−0.489 − 0.103i)4-s + (−0.0467 − 0.444i)5-s + (−0.702 − 0.0790i)6-s + (0.786 − 0.873i)7-s + (0.109 − 0.336i)8-s + (−0.999 − 0.0145i)9-s + 0.316·10-s + (0.275 + 0.961i)11-s + (0.107 − 0.488i)12-s + (−1.13 + 0.503i)13-s + (0.556 + 0.617i)14-s + (0.445 − 0.0435i)15-s + (0.228 + 0.101i)16-s + (−0.849 + 0.616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.101095 - 0.795162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.101095 - 0.795162i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (0.0126 - 1.73i)T \) |
| 5 | \( 1 + (0.104 + 0.994i)T \) |
| 11 | \( 1 + (-0.912 - 3.18i)T \) |
| good | 7 | \( 1 + (-2.08 + 2.31i)T + (-0.731 - 6.96i)T^{2} \) |
| 13 | \( 1 + (4.07 - 1.81i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (3.50 - 2.54i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.484 + 1.48i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.35 - 4.84i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (9.63 - 4.29i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.85 - 5.72i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.09 - 3.43i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-3.01 - 5.22i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.49 + 1.16i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-5.80 - 4.21i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (10.4 + 2.21i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (8.57 + 3.81i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-1.11 + 1.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.31 + 3.13i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.11 + 12.6i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.55 + 14.8i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-7.12 - 3.17i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 - 8.41T + 89T^{2} \) |
| 97 | \( 1 + (-1.06 + 10.1i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−10.35243320022412640350229029191, −9.284117865213017605594290375892, −9.092688044128605001164529315582, −7.75279028281528464838292927019, −7.31844197674250898403525722363, −6.10809100905532902403541662459, −4.82102787669334942982886900687, −4.67702773905546659911322800094, −3.65828600967457736253387549715, −1.79696325130348336899798598707,
0.36039748307218675990695263481, 2.12405020635769311250001851927, 2.57179576705324686670033407847, 3.96084421705365175160331792397, 5.38071600788164779902664861601, 5.92855165665404855192987852487, 7.23253456402423841066570675760, 7.86918695961219305433767796156, 8.768037689665532924914196515445, 9.374369435093546877136096880271
