| L(s) = 1 | + (−85.4 − 147. i)2-s + (275. − 477. i)3-s + (−1.05e4 + 1.81e4i)4-s + (1.84e4 + 3.19e4i)5-s − 9.41e4·6-s + (−2.96e5 − 9.57e4i)7-s + 2.18e6·8-s + (6.45e5 + 1.11e6i)9-s + (3.15e6 − 5.46e6i)10-s + (−4.17e5 + 7.23e5i)11-s + (5.78e6 + 1.00e7i)12-s − 1.14e7·13-s + (1.11e7 + 5.20e7i)14-s + 2.03e7·15-s + (−1.01e8 − 1.75e8i)16-s + (−1.80e7 + 3.12e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.943 − 1.63i)2-s + (0.218 − 0.377i)3-s + (−1.28 + 2.22i)4-s + (0.528 + 0.915i)5-s − 0.823·6-s + (−0.951 − 0.307i)7-s + 2.95·8-s + (0.404 + 0.701i)9-s + (0.997 − 1.72i)10-s + (−0.0710 + 0.123i)11-s + (0.559 + 0.969i)12-s − 0.656·13-s + (0.395 + 1.84i)14-s + 0.461·15-s + (−1.50 − 2.60i)16-s + (−0.181 + 0.313i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.700238 + 0.0517448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.700238 + 0.0517448i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.96e5 + 9.57e4i)T \) |
| good | 2 | \( 1 + (85.4 + 147. i)T + (-4.09e3 + 7.09e3i)T^{2} \) |
| 3 | \( 1 + (-275. + 477. i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-1.84e4 - 3.19e4i)T + (-6.10e8 + 1.05e9i)T^{2} \) |
| 11 | \( 1 + (4.17e5 - 7.23e5i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 13 | \( 1 + 1.14e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + (1.80e7 - 3.12e7i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (-2.00e8 - 3.46e8i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-2.53e8 - 4.39e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 + 4.26e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (1.81e9 - 3.15e9i)T + (-1.22e19 - 2.11e19i)T^{2} \) |
| 37 | \( 1 + (-2.77e8 - 4.80e8i)T + (-1.21e20 + 2.10e20i)T^{2} \) |
| 41 | \( 1 + 3.81e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 7.34e8T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-1.79e10 - 3.10e10i)T + (-2.73e21 + 4.72e21i)T^{2} \) |
| 53 | \( 1 + (7.30e10 - 1.26e11i)T + (-1.30e22 - 2.25e22i)T^{2} \) |
| 59 | \( 1 + (1.41e11 - 2.45e11i)T + (-5.24e22 - 9.09e22i)T^{2} \) |
| 61 | \( 1 + (1.37e11 + 2.37e11i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-4.52e11 + 7.83e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 - 1.65e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + (-3.13e11 + 5.42e11i)T + (-8.35e23 - 1.44e24i)T^{2} \) |
| 79 | \( 1 + (3.45e11 + 5.98e11i)T + (-2.33e24 + 4.04e24i)T^{2} \) |
| 83 | \( 1 - 3.52e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (6.89e11 + 1.19e12i)T + (-1.09e25 + 1.90e25i)T^{2} \) |
| 97 | \( 1 - 3.29e10T + 6.73e25T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.06514144157477402804808424660, −18.33569746241086041783125484403, −16.77385927492459937250234506460, −13.75081908340540079793235780639, −12.47509556685131206941760071039, −10.60768761278136901079327045255, −9.684329555522509861816301230558, −7.50591331249019491460899909862, −3.26446198875050889399143504359, −1.77903805683052652634456281476,
0.50317579191205583386870695820, 5.18815143019989961465218125582, 6.89540107246294267078783483304, 9.046713470563806736503894712132, 9.633309710744558840231068637828, 13.24286994934938135623132226164, 15.08033604303748346303013611813, 16.11599699119194942688986533202, 17.21350901498642923372799179337, 18.56727426953711190245850907436
