L(s) = 1 | + (3.90 + 6.76i)2-s + (−11.7 + 20.3i)3-s + (−14.5 + 25.1i)4-s + (37.1 + 64.3i)5-s − 183.·6-s + 22.8·8-s + (−155. − 268. i)9-s + (−290. + 502. i)10-s + (212. − 367. i)11-s + (−341. − 592. i)12-s − 252.·13-s − 1.74e3·15-s + (554. + 960. i)16-s + (552. − 956. i)17-s + (1.21e3 − 2.09e3i)18-s + (3.23 + 5.60i)19-s + ⋯ |
L(s) = 1 | + (0.690 + 1.19i)2-s + (−0.754 + 1.30i)3-s + (−0.454 + 0.786i)4-s + (0.664 + 1.15i)5-s − 2.08·6-s + 0.126·8-s + (−0.638 − 1.10i)9-s + (−0.917 + 1.58i)10-s + (0.528 − 0.915i)11-s + (−0.685 − 1.18i)12-s − 0.413·13-s − 2.00·15-s + (0.541 + 0.937i)16-s + (0.463 − 0.802i)17-s + (0.881 − 1.52i)18-s + (0.00205 + 0.00356i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.331300 - 2.02573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.331300 - 2.02573i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (-3.90 - 6.76i)T + (-16 + 27.7i)T^{2} \) |
| 3 | \( 1 + (11.7 - 20.3i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-37.1 - 64.3i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-212. + 367. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 252.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-552. + 956. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-3.23 - 5.60i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.80e3 - 3.12e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 5.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (1.41e3 - 2.44e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-1.02e3 - 1.77e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 9.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (8.51e3 + 1.47e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.97e4 + 3.42e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.69e4 - 2.94e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.41e4 + 2.45e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.80e4 - 4.85e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 1.55e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.91e4 + 6.77e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.26e4 - 3.92e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 - 1.38e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-3.44e4 - 5.96e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.08e5T + 8.58e9T^{2} \) |
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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
−15.15847840265542252895025179137, −14.48329159651383931637094889700, −13.52038704888575235264212918300, −11.45949511801910578425509376074, −10.54614940909286375196430180052, −9.399210986151361845546463598291, −7.24420777200368462948541240464, −6.01900307272834309488203636124, −5.18780835087480420964670398745, −3.55882501981732478680781781255,
1.03031907024905714493777001066, 2.04075490907682545513325787181, 4.55618893221208174406971048281, 5.87360222460017352265734089575, 7.49889964007213940707466960142, 9.381993051123879888130872564287, 10.85990767642170353774269852147, 12.17877900957590538010028864274, 12.60869044177866024989671828475, 13.22827555724588715841668644051