| L(s) = 1 | + (1.30 + 0.540i)2-s + (0.755 − 0.654i)3-s + (1.41 + 1.41i)4-s + (0.486 − 0.0699i)5-s + (1.34 − 0.446i)6-s + (−1.63 + 3.58i)7-s + (1.08 + 2.61i)8-s + (0.142 − 0.989i)9-s + (0.673 + 0.171i)10-s + (−0.457 + 1.55i)11-s + (1.99 + 0.141i)12-s + (5.75 − 2.62i)13-s + (−4.08 + 3.80i)14-s + (0.321 − 0.371i)15-s + (0.00298 + 3.99i)16-s + (−2.29 − 1.47i)17-s + ⋯ |
| L(s) = 1 | + (0.923 + 0.382i)2-s + (0.436 − 0.378i)3-s + (0.707 + 0.706i)4-s + (0.217 − 0.0312i)5-s + (0.547 − 0.182i)6-s + (−0.619 + 1.35i)7-s + (0.383 + 0.923i)8-s + (0.0474 − 0.329i)9-s + (0.212 + 0.0543i)10-s + (−0.138 + 0.470i)11-s + (0.575 + 0.0409i)12-s + (1.59 − 0.729i)13-s + (−1.09 + 1.01i)14-s + (0.0831 − 0.0959i)15-s + (0.000746 + 0.999i)16-s + (−0.556 − 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.59237 + 1.21655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.59237 + 1.21655i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 - 0.540i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 23 | \( 1 + (0.305 - 4.78i)T \) |
| good | 5 | \( 1 + (-0.486 + 0.0699i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.63 - 3.58i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.457 - 1.55i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-5.75 + 2.62i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (2.29 + 1.47i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (1.64 + 2.55i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.38 + 5.26i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-5.90 + 6.81i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.882 + 0.126i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.06 + 7.41i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-4.02 + 3.48i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 2.79T + 47T^{2} \) |
| 53 | \( 1 + (12.9 + 5.92i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (2.39 - 1.09i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-10.0 - 8.73i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.39 - 4.75i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (10.1 - 2.98i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (10.7 - 6.93i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.29 + 2.84i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.214 - 0.0308i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (4.15 + 4.79i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.12 + 7.80i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.30257078069631862241033837287, −9.943397515239081615715821586845, −8.879016096172071767183827136156, −8.229468176268150990376870047107, −7.14744069349610809157015608125, −6.05067002331700817239421275930, −5.69797937148662701494189075985, −4.20284827924871514058436541270, −3.03305153681429407295671927674, −2.13909635746306841242154530405,
1.40951951547950660570964833092, 3.06597067676869420172871879909, 3.90996044430172330807098920018, 4.62365729739705558342045106447, 6.26910298277932051387627242549, 6.56255797808485495853793123534, 7.991564135973811206647471325539, 9.029446119501382657022009853198, 10.21570044786770482133498330151, 10.59166716665681703533448418720
