L(s) = 1 | + 38.4i·2-s + (−824. − 2.02e3i)3-s + 1.49e4·4-s − 1.22e5i·5-s + (7.79e4 − 3.17e4i)6-s − 6.55e4·7-s + 1.20e6i·8-s + (−3.42e6 + 3.34e6i)9-s + 4.73e6·10-s − 6.41e6i·11-s + (−1.22e7 − 3.01e7i)12-s + 6.71e7·13-s − 2.52e6i·14-s + (−2.49e8 + 1.01e8i)15-s + 1.97e8·16-s + 3.10e8i·17-s + ⋯ |
L(s) = 1 | + 0.300i·2-s + (−0.377 − 0.926i)3-s + 0.909·4-s − 1.57i·5-s + (0.278 − 0.113i)6-s − 0.0796·7-s + 0.574i·8-s + (−0.715 + 0.698i)9-s + 0.473·10-s − 0.329i·11-s + (−0.342 − 0.842i)12-s + 1.06·13-s − 0.0239i·14-s + (−1.45 + 0.593i)15-s + 0.736·16-s + 0.756i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.926i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.377 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(1.24589 - 0.837950i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24589 - 0.837950i\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (824. + 2.02e3i)T \) |
good | 2 | \( 1 - 38.4iT - 1.63e4T^{2} \) |
| 5 | \( 1 + 1.22e5iT - 6.10e9T^{2} \) |
| 7 | \( 1 + 6.55e4T + 6.78e11T^{2} \) |
| 11 | \( 1 + 6.41e6iT - 3.79e14T^{2} \) |
| 13 | \( 1 - 6.71e7T + 3.93e15T^{2} \) |
| 17 | \( 1 - 3.10e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 2.64e8T + 7.99e17T^{2} \) |
| 23 | \( 1 - 3.74e9iT - 1.15e19T^{2} \) |
| 29 | \( 1 + 2.30e10iT - 2.97e20T^{2} \) |
| 31 | \( 1 - 1.42e10T + 7.56e20T^{2} \) |
| 37 | \( 1 - 7.16e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + 9.96e10iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 2.62e11T + 7.38e22T^{2} \) |
| 47 | \( 1 - 5.94e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 7.14e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 - 2.54e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 2.14e12T + 9.87e24T^{2} \) |
| 67 | \( 1 + 3.16e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 4.44e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 + 1.16e13T + 1.22e26T^{2} \) |
| 79 | \( 1 + 1.15e13T + 3.68e26T^{2} \) |
| 83 | \( 1 - 1.72e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 9.51e12iT - 1.95e27T^{2} \) |
| 97 | \( 1 - 8.81e13T + 6.52e27T^{2} \) |
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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
−23.55981909766658428806926601903, −20.92567164050193955648296971203, −19.53947478625024875370539018875, −17.25753596520518826083979307961, −16.03987864387926298373883356313, −13.20507597558205478365606358569, −11.63792977795387934645811014307, −8.165945495526480065346542857443, −5.88458104682680655033135563454, −1.31128672141184471137906988743,
3.12161474398950416375773634813, 6.56545316065626169403218547247, 10.31822612198311257777724444974, 11.42137507976158597699455246577, 14.74763793859724009735873919564, 16.10481448291544310408982676059, 18.36119711644519544387469024253, 20.36681019448452607496355958083, 21.82885815598797112380681122164, 23.06363182485434568961348141733