L(s) = 1 | + (7.66 − 25.8i)3-s + (−37.2 − 119. i)5-s − 46.7i·7-s + (−611. − 396. i)9-s − 448. i·11-s + 2.07e3i·13-s + (−3.37e3 + 50.5i)15-s − 5.98e3·17-s + 7.40e3·19-s + (−1.20e3 − 358. i)21-s − 1.71e4·23-s + (−1.28e4 + 8.89e3i)25-s + (−1.49e4 + 1.27e4i)27-s − 3.75e4i·29-s − 1.79e4·31-s + ⋯ |
L(s) = 1 | + (0.283 − 0.958i)3-s + (−0.298 − 0.954i)5-s − 0.136i·7-s + (−0.838 − 0.544i)9-s − 0.337i·11-s + 0.943i·13-s + (−0.999 + 0.0149i)15-s − 1.21·17-s + 1.07·19-s + (−0.130 − 0.0386i)21-s − 1.41·23-s + (−0.822 + 0.569i)25-s + (−0.759 + 0.649i)27-s − 1.53i·29-s − 0.603·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0149i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 + 0.0149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.00799827 - 1.06822i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00799827 - 1.06822i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-7.66 + 25.8i)T \) |
| 5 | \( 1 + (37.2 + 119. i)T \) |
good | 7 | \( 1 + 46.7iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 448. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.07e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 5.98e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 7.40e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.71e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 3.75e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.79e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.62e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.28e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 2.50e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 5.18e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.70e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.18e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 3.83e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.59e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 4.52e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 7.84e4iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 4.10e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 6.30e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.45e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.76e6iT - 8.32e11T^{2} \) |
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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
−13.38163872687343535327097275413, −12.17476604245462695254654437245, −11.41860593099302284715245466415, −9.423851263053981566914977825963, −8.439245356617273974504397279804, −7.30459249351420387907840189182, −5.86074041549824513210582859724, −4.07781222963225949092421570834, −1.99330291030442800136958002334, −0.40983004168409188473700010386,
2.64927072104145243425493137691, 3.94544963115327861372820595598, 5.56134594513863441653586083439, 7.24846506470169002873502422418, 8.596922243214156232206296358727, 9.968795967667939935566300821634, 10.77735819528588490158590111097, 11.90039561658199465198573245539, 13.56368511052505057051623424793, 14.61682834479970137571436827837