| L(s) = 1 | + (−4.33 + 1.27i)2-s + (10.2 + 11.7i)3-s + (−36.6 + 23.5i)4-s + (168. − 24.2i)5-s + (−59.2 − 38.1i)6-s + (49.4 + 22.5i)7-s + (318. − 367. i)8-s + (−34.5 + 240. i)9-s + (−700. + 319. i)10-s + (579. − 1.97e3i)11-s + (−651. − 191. i)12-s + (484. + 1.06e3i)13-s + (−243. − 34.9i)14-s + (2.00e3 + 1.73e3i)15-s + (244. − 535. i)16-s + (84.9 − 132. i)17-s + ⋯ |
| L(s) = 1 | + (−0.542 + 0.159i)2-s + (0.378 + 0.436i)3-s + (−0.572 + 0.367i)4-s + (1.34 − 0.193i)5-s + (−0.274 − 0.176i)6-s + (0.144 + 0.0658i)7-s + (0.622 − 0.717i)8-s + (−0.0474 + 0.329i)9-s + (−0.700 + 0.319i)10-s + (0.435 − 1.48i)11-s + (−0.376 − 0.110i)12-s + (0.220 + 0.482i)13-s + (−0.0886 − 0.0127i)14-s + (0.594 + 0.514i)15-s + (0.0596 − 0.130i)16-s + (0.0172 − 0.0269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.618i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.786 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.73978 + 0.602164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73978 + 0.602164i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-10.2 - 11.7i)T \) |
| 23 | \( 1 + (-1.21e4 - 366. i)T \) |
| good | 2 | \( 1 + (4.33 - 1.27i)T + (53.8 - 34.6i)T^{2} \) |
| 5 | \( 1 + (-168. + 24.2i)T + (1.49e4 - 4.40e3i)T^{2} \) |
| 7 | \( 1 + (-49.4 - 22.5i)T + (7.70e4 + 8.89e4i)T^{2} \) |
| 11 | \( 1 + (-579. + 1.97e3i)T + (-1.49e6 - 9.57e5i)T^{2} \) |
| 13 | \( 1 + (-484. - 1.06e3i)T + (-3.16e6 + 3.64e6i)T^{2} \) |
| 17 | \( 1 + (-84.9 + 132. i)T + (-1.00e7 - 2.19e7i)T^{2} \) |
| 19 | \( 1 + (-1.87e3 - 2.91e3i)T + (-1.95e7 + 4.27e7i)T^{2} \) |
| 29 | \( 1 + (-3.27e4 - 2.10e4i)T + (2.47e8 + 5.41e8i)T^{2} \) |
| 31 | \( 1 + (8.92e3 - 1.02e4i)T + (-1.26e8 - 8.78e8i)T^{2} \) |
| 37 | \( 1 + (-4.66e4 - 6.70e3i)T + (2.46e9 + 7.22e8i)T^{2} \) |
| 41 | \( 1 + (-1.71e4 - 1.19e5i)T + (-4.55e9 + 1.33e9i)T^{2} \) |
| 43 | \( 1 + (-4.82e4 + 4.17e4i)T + (8.99e8 - 6.25e9i)T^{2} \) |
| 47 | \( 1 + 2.22e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (2.01e5 + 9.20e4i)T + (1.45e10 + 1.67e10i)T^{2} \) |
| 59 | \( 1 + (1.07e5 + 2.34e5i)T + (-2.76e10 + 3.18e10i)T^{2} \) |
| 61 | \( 1 + (1.78e5 + 1.54e5i)T + (7.33e9 + 5.09e10i)T^{2} \) |
| 67 | \( 1 + (-1.53e5 - 5.21e5i)T + (-7.60e10 + 4.89e10i)T^{2} \) |
| 71 | \( 1 + (-5.02e4 + 1.47e4i)T + (1.07e11 - 6.92e10i)T^{2} \) |
| 73 | \( 1 + (-2.21e5 + 1.42e5i)T + (6.28e10 - 1.37e11i)T^{2} \) |
| 79 | \( 1 + (-1.74e5 + 7.95e4i)T + (1.59e11 - 1.83e11i)T^{2} \) |
| 83 | \( 1 + (-6.96e5 - 1.00e5i)T + (3.13e11 + 9.21e10i)T^{2} \) |
| 89 | \( 1 + (4.43e5 - 3.84e5i)T + (7.07e10 - 4.91e11i)T^{2} \) |
| 97 | \( 1 + (-1.87e5 + 2.69e4i)T + (7.99e11 - 2.34e11i)T^{2} \) |
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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
−13.79405595891051145247328133353, −12.85244732812795486279799292166, −11.10061311257292157132998541822, −9.822633970429523861311522759410, −9.069550261504414173342894219940, −8.235600915350557778731165381855, −6.39741018297906420428314280529, −4.94740687157913831080585620293, −3.25974432014494113654870064922, −1.20272427871027690751868747816,
1.13909043619101800790652789471, 2.33020975979416159322056470197, 4.73251059864563004376423047053, 6.17279852825273302284523859480, 7.61930080359871517223523102362, 9.123094039701902946579259906986, 9.725660071941400226943968657573, 10.77762241544875337181758814218, 12.53000060616820606584679394771, 13.54178141651015408417207255366
