| L(s) = 1 | + (80.7 + 139. i)2-s + (810. − 1.40e3i)3-s + (−8.96e3 + 1.55e4i)4-s + (1.89e4 + 3.27e4i)5-s + 2.61e5·6-s + (−4.02e4 + 3.08e5i)7-s − 1.57e6·8-s + (−5.17e5 − 8.95e5i)9-s + (−3.05e6 + 5.29e6i)10-s + (4.43e6 − 7.69e6i)11-s + (1.45e7 + 2.51e7i)12-s − 7.42e6·13-s + (−4.64e7 + 1.93e7i)14-s + 6.13e7·15-s + (−5.36e7 − 9.28e7i)16-s + (5.51e7 − 9.55e7i)17-s + ⋯ |
| L(s) = 1 | + (0.892 + 1.54i)2-s + (0.641 − 1.11i)3-s + (−1.09 + 1.89i)4-s + (0.541 + 0.938i)5-s + 2.29·6-s + (−0.129 + 0.991i)7-s − 2.12·8-s + (−0.324 − 0.561i)9-s + (−0.967 + 1.67i)10-s + (0.755 − 1.30i)11-s + (1.40 + 2.43i)12-s − 0.426·13-s + (−1.64 + 0.685i)14-s + 1.39·15-s + (−0.798 − 1.38i)16-s + (0.554 − 0.959i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.80178 + 2.52344i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80178 + 2.52344i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (4.02e4 - 3.08e5i)T \) |
| good | 2 | \( 1 + (-80.7 - 139. i)T + (-4.09e3 + 7.09e3i)T^{2} \) |
| 3 | \( 1 + (-810. + 1.40e3i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-1.89e4 - 3.27e4i)T + (-6.10e8 + 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-4.43e6 + 7.69e6i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 13 | \( 1 + 7.42e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-5.51e7 + 9.55e7i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (-2.40e7 - 4.16e7i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (3.77e8 + 6.54e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 - 1.64e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (1.67e9 - 2.90e9i)T + (-1.22e19 - 2.11e19i)T^{2} \) |
| 37 | \( 1 + (1.14e10 + 1.97e10i)T + (-1.21e20 + 2.10e20i)T^{2} \) |
| 41 | \( 1 + 2.09e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.50e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-3.75e10 - 6.50e10i)T + (-2.73e21 + 4.72e21i)T^{2} \) |
| 53 | \( 1 + (-2.75e9 + 4.77e9i)T + (-1.30e22 - 2.25e22i)T^{2} \) |
| 59 | \( 1 + (4.28e10 - 7.41e10i)T + (-5.24e22 - 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-4.21e9 - 7.30e9i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (6.44e11 - 1.11e12i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 - 5.15e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + (3.29e11 - 5.70e11i)T + (-8.35e23 - 1.44e24i)T^{2} \) |
| 79 | \( 1 + (-6.26e11 - 1.08e12i)T + (-2.33e24 + 4.04e24i)T^{2} \) |
| 83 | \( 1 + 2.68e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (2.39e12 + 4.15e12i)T + (-1.09e25 + 1.90e25i)T^{2} \) |
| 97 | \( 1 + 2.40e11T + 6.73e25T^{2} \) |
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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
−19.00906231715598113009032624956, −17.99007896718655087180068829911, −16.19033763956194989976329681667, −14.39983825010378969624194546744, −13.98106797838035507773641061965, −12.36605270901181917707072833017, −8.598885970312412975411516055434, −7.03243564161455914448412559819, −5.88835288778541733710037157374, −2.87051061910312746608041678932,
1.52300210422411704533792068226, 3.72883779888362490663512794764, 4.78624202477342760768191943460, 9.470429525731233334130715987924, 10.22713921743504978520242033141, 12.29827866989666319891746095610, 13.67895169225928665318485613660, 14.94921552949794610068089938425, 17.12141994810270255918200002757, 19.74521545391327661657038271022
