| L(s) = 1 | + (−22.3 + 3.94i)3-s + (18.7 − 15.7i)5-s + (52.4 − 90.7i)7-s + (−200. + 73.1i)9-s + (−640. − 1.10e3i)11-s + (888. + 156. i)13-s + (−357. + 425. i)15-s + (3.79e3 + 1.37e3i)17-s + (983. + 6.78e3i)19-s + (−813. + 2.23e3i)21-s + (1.22e4 + 1.02e4i)23-s + (−2.60e3 + 1.47e4i)25-s + (1.85e4 − 1.06e4i)27-s + (9.84e3 + 2.70e4i)29-s + (137. + 79.5i)31-s + ⋯ |
| L(s) = 1 | + (−0.827 + 0.145i)3-s + (0.150 − 0.125i)5-s + (0.152 − 0.264i)7-s + (−0.275 + 0.100i)9-s + (−0.481 − 0.833i)11-s + (0.404 + 0.0712i)13-s + (−0.105 + 0.126i)15-s + (0.771 + 0.280i)17-s + (0.143 + 0.989i)19-s + (−0.0878 + 0.241i)21-s + (1.00 + 0.841i)23-s + (−0.166 + 0.947i)25-s + (0.941 − 0.543i)27-s + (0.403 + 1.10i)29-s + (0.00462 + 0.00267i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.08307 + 0.520588i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08307 + 0.520588i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-983. - 6.78e3i)T \) |
| good | 3 | \( 1 + (22.3 - 3.94i)T + (685. - 249. i)T^{2} \) |
| 5 | \( 1 + (-18.7 + 15.7i)T + (2.71e3 - 1.53e4i)T^{2} \) |
| 7 | \( 1 + (-52.4 + 90.7i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (640. + 1.10e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-888. - 156. i)T + (4.53e6 + 1.65e6i)T^{2} \) |
| 17 | \( 1 + (-3.79e3 - 1.37e3i)T + (1.84e7 + 1.55e7i)T^{2} \) |
| 23 | \( 1 + (-1.22e4 - 1.02e4i)T + (2.57e7 + 1.45e8i)T^{2} \) |
| 29 | \( 1 + (-9.84e3 - 2.70e4i)T + (-4.55e8 + 3.82e8i)T^{2} \) |
| 31 | \( 1 + (-137. - 79.5i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 1.13e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (9.89e3 - 1.74e3i)T + (4.46e9 - 1.62e9i)T^{2} \) |
| 43 | \( 1 + (-5.06e4 + 4.25e4i)T + (1.09e9 - 6.22e9i)T^{2} \) |
| 47 | \( 1 + (-5.47e4 + 1.99e4i)T + (8.25e9 - 6.92e9i)T^{2} \) |
| 53 | \( 1 + (7.87e4 - 9.38e4i)T + (-3.84e9 - 2.18e10i)T^{2} \) |
| 59 | \( 1 + (-2.09e4 + 5.75e4i)T + (-3.23e10 - 2.71e10i)T^{2} \) |
| 61 | \( 1 + (-1.04e5 - 8.73e4i)T + (8.94e9 + 5.07e10i)T^{2} \) |
| 67 | \( 1 + (7.68e4 + 2.11e5i)T + (-6.92e10 + 5.81e10i)T^{2} \) |
| 71 | \( 1 + (1.44e5 + 1.71e5i)T + (-2.22e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (4.44e4 + 2.51e5i)T + (-1.42e11 + 5.17e10i)T^{2} \) |
| 79 | \( 1 + (2.81e5 - 4.97e4i)T + (2.28e11 - 8.31e10i)T^{2} \) |
| 83 | \( 1 + (-8.08e4 + 1.40e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-5.92e5 - 1.04e5i)T + (4.67e11 + 1.69e11i)T^{2} \) |
| 97 | \( 1 + (2.81e5 - 7.73e5i)T + (-6.38e11 - 5.35e11i)T^{2} \) |
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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
−13.44620910453267793722440025620, −12.22187754167780286023297021060, −11.14140341544495987582443570301, −10.41976675771787101719924348984, −8.905901415818226923068573576875, −7.62840639856745864222506146252, −6.00019697753396852832623469992, −5.18125469823932514857009120146, −3.35643842754821637645659648248, −1.11326542857207675399353350620,
0.62310822821897591150731564312, 2.63831847875826155070801206501, 4.73099530946119375098228857556, 5.88498809406413972468857925669, 7.07679167218203645743721277424, 8.540890304424077427927275869529, 9.934966285302835917022850732955, 11.04848629598976735504828486700, 11.99033381672396626343386026253, 12.93859430057060584143174441543
