| 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
−19.74521545391327661657038271022, −17.12141994810270255918200002757, −14.94921552949794610068089938425, −13.67895169225928665318485613660, −12.29827866989666319891746095610, −10.22713921743504978520242033141, −9.470429525731233334130715987924, −4.78624202477342760768191943460, −3.72883779888362490663512794764, −1.52300210422411704533792068226,
2.87051061910312746608041678932, 5.88835288778541733710037157374, 7.03243564161455914448412559819, 8.598885970312412975411516055434, 12.36605270901181917707072833017, 13.98106797838035507773641061965, 14.39983825010378969624194546744, 16.19033763956194989976329681667, 17.99007896718655087180068829911, 19.00906231715598113009032624956
