| L(s) = 1 | + (1.36 + 0.366i)2-s + (2.81 − 1.62i)3-s + (1.73 + i)4-s + (−1.01 + 4.89i)5-s + (4.44 − 1.19i)6-s + (3.91 − 1.04i)7-s + (1.99 + 2i)8-s + (0.800 − 1.38i)9-s + (−3.17 + 6.31i)10-s + (−3.41 − 0.915i)11-s + 6.51·12-s + (−1.34 − 12.9i)13-s + 5.72·14-s + (5.10 + 15.4i)15-s + (1.99 + 3.46i)16-s + (14.4 − 24.9i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (0.939 − 0.542i)3-s + (0.433 + 0.250i)4-s + (−0.203 + 0.979i)5-s + (0.741 − 0.198i)6-s + (0.559 − 0.149i)7-s + (0.249 + 0.250i)8-s + (0.0889 − 0.153i)9-s + (−0.317 + 0.631i)10-s + (−0.310 − 0.0832i)11-s + 0.542·12-s + (−0.103 − 0.994i)13-s + 0.409·14-s + (0.340 + 1.03i)15-s + (0.124 + 0.216i)16-s + (0.848 − 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.57405 + 0.257392i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.57405 + 0.257392i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 5 | \( 1 + (1.01 - 4.89i)T \) |
| 13 | \( 1 + (1.34 + 12.9i)T \) |
| good | 3 | \( 1 + (-2.81 + 1.62i)T + (4.5 - 7.79i)T^{2} \) |
| 7 | \( 1 + (-3.91 + 1.04i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (3.41 + 0.915i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (-14.4 + 24.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (35.3 - 9.48i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (2.41 + 4.18i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-16.0 - 27.8i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (5.98 + 5.98i)T + 961iT^{2} \) |
| 37 | \( 1 + (10.3 - 38.5i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (-4.98 + 18.6i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (-9.17 + 15.8i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (32.9 + 32.9i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + 45.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-20.3 - 75.8i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-36.9 + 63.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (22.4 + 6.02i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-94.9 + 25.4i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-71.9 - 71.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 3.01T + 6.24e3T^{2} \) |
| 83 | \( 1 + (7.10 - 7.10i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-28.7 - 7.70i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-18.2 - 68.0i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−13.32347472822716670560851650366, −12.33812070717070529741528237075, −11.11965307072217919685136967062, −10.21086446201892325099297056235, −8.379562099548392262896402294816, −7.70307593331470120481759909975, −6.70503758646402656204186653605, −5.16036344564202391189337510255, −3.41559747218392937507427361223, −2.36523585512167534222809152959,
2.04148076192506098536453696973, 3.86488139218410257211762296395, 4.65357474360126198740466261960, 6.15494250309326743653225161183, 8.010525035714347824318597782981, 8.733185398836237064059581760741, 9.839305738261889929821958956076, 11.10079929380458055545010115113, 12.25712349649086709711589585925, 13.02012862799534385361914982178
