| L(s) = 1 | + (11.1 + 11.1i)2-s − 29.5i·3-s + 186. i·4-s + (−126. + 126. i)5-s + (331. − 331. i)6-s + 249.·7-s + (−1.37e3 + 1.37e3i)8-s − 146.·9-s − 2.84e3·10-s + 1.25e3i·11-s + 5.52e3·12-s + (1.26e3 − 1.26e3i)13-s + (2.79e3 + 2.79e3i)14-s + (3.75e3 + 3.75e3i)15-s − 1.88e4·16-s + (3.46e3 − 3.46e3i)17-s + ⋯ |
| L(s) = 1 | + (1.39 + 1.39i)2-s − 1.09i·3-s + 2.91i·4-s + (−1.01 + 1.01i)5-s + (1.53 − 1.53i)6-s + 0.727·7-s + (−2.68 + 2.68i)8-s − 0.200·9-s − 2.84·10-s + 0.943i·11-s + 3.19·12-s + (0.575 − 0.575i)13-s + (1.01 + 1.01i)14-s + (1.11 + 1.11i)15-s − 4.60·16-s + (0.705 − 0.705i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.713i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.701 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.16119 + 2.77015i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16119 + 2.77015i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-5.04e4 - 4.60e3i)T \) |
| good | 2 | \( 1 + (-11.1 - 11.1i)T + 64iT^{2} \) |
| 3 | \( 1 + 29.5iT - 729T^{2} \) |
| 5 | \( 1 + (126. - 126. i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 - 249.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.25e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + (-1.26e3 + 1.26e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + (-3.46e3 + 3.46e3i)T - 2.41e7iT^{2} \) |
| 19 | \( 1 + (-2.76e3 + 2.76e3i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 + (-3.78e3 + 3.78e3i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 + (-2.14e4 - 2.14e4i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 + (2.40e3 + 2.40e3i)T + 8.87e8iT^{2} \) |
| 41 | \( 1 - 5.50e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (7.67e4 - 7.67e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 2.73e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 1.86e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + (-5.52e4 + 5.52e4i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (9.69e4 + 9.69e4i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 + 3.54e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 3.67e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 4.94e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + (7.78e4 - 7.78e4i)T - 2.43e11iT^{2} \) |
| 83 | \( 1 - 2.74e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-5.19e4 - 5.19e4i)T + 4.96e11iT^{2} \) |
| 97 | \( 1 + (-3.67e5 + 3.67e5i)T - 8.32e11iT^{2} \) |
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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
−15.16878469033983194533014283620, −14.53327665294225099476936585729, −13.38802888501982725848341630318, −12.31133443122541069888821245212, −11.44901212691018226289961674483, −8.043420169704730029615936537558, −7.41967060148024926252936654731, −6.54473717189443065222073457194, −4.74348336189332215777852927003, −3.06166810591880702546010947025,
1.11853798220628370266396714433, 3.63374063486876082568781382022, 4.39389097480250616703983395542, 5.53137466286780845116553675301, 8.729074973008504339593883093736, 10.17595505255879542358439561297, 11.30723814481027390683961398870, 11.99859729413534800690181987380, 13.30936865944240240545438552314, 14.51707578228814098082795582231
