| L(s) = 1 | + (12.8 + 44.9i)3-s + (156. − 270. i)5-s + (626. + 1.08e3i)7-s + (−1.85e3 + 1.15e3i)9-s + (992. + 1.71e3i)11-s + (6.03e3 − 1.04e4i)13-s + (1.41e4 + 3.56e3i)15-s − 5.45e3·17-s + 5.19e4·19-s + (−4.07e4 + 4.20e4i)21-s + (−4.55e4 + 7.89e4i)23-s + (−9.81e3 − 1.69e4i)25-s + (−7.56e4 − 6.88e4i)27-s + (5.63e4 + 9.76e4i)29-s + (−1.65e5 + 2.87e5i)31-s + ⋯ |
| L(s) = 1 | + (0.273 + 0.961i)3-s + (0.559 − 0.968i)5-s + (0.689 + 1.19i)7-s + (−0.850 + 0.526i)9-s + (0.224 + 0.389i)11-s + (0.762 − 1.32i)13-s + (1.08 + 0.272i)15-s − 0.269·17-s + 1.73·19-s + (−0.960 + 0.990i)21-s + (−0.780 + 1.35i)23-s + (−0.125 − 0.217i)25-s + (−0.739 − 0.673i)27-s + (0.429 + 0.743i)29-s + (−0.999 + 1.73i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.03493 + 1.47241i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03493 + 1.47241i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-12.8 - 44.9i)T \) |
| good | 5 | \( 1 + (-156. + 270. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-626. - 1.08e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-992. - 1.71e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-6.03e3 + 1.04e4i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + 5.45e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.19e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (4.55e4 - 7.89e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-5.63e4 - 9.76e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.65e5 - 2.87e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 3.13e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (9.59e4 - 1.66e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-8.43e4 - 1.46e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (2.40e5 + 4.16e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 5.62e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-5.23e5 + 9.06e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-5.87e5 - 1.01e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.63e6 + 2.82e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 2.19e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.66e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (1.01e6 + 1.76e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.79e6 + 3.11e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 5.04e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-4.80e6 - 8.33e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
| 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
−13.48564469600405284434196232791, −12.26730939580027508536358241759, −11.16163235534681077066109212669, −9.761287275110960456580468928110, −8.963035752816794909499571707562, −8.035042915814927414609931149949, −5.54614107582722952641585295654, −5.11627729171235834450230051881, −3.25243090316045374521075791598, −1.50918998680055006695783418570,
0.943563004945833440983450741995, 2.29426999052282543958733640167, 3.95586962943106782286703360966, 6.11989963010901554633826775317, 7.01232674044205012641972864975, 8.059629492377333445744708078866, 9.551974308180845308569666386955, 10.96552553089484221981595431284, 11.67760143058650004480459952593, 13.37631490720647254632221334144
