| L(s) = 1 | + (−4.57 − 3.32i)2-s + (14.1 − 24.5i)3-s + (9.95 + 30.4i)4-s + (24.5 − 42.4i)5-s + (−146. + 65.4i)6-s + 6.61i·7-s + (55.4 − 172. i)8-s + (−281. − 487. i)9-s + (−253. + 113. i)10-s − 102. i·11-s + (888. + 186. i)12-s + (718. − 415. i)13-s + (21.9 − 30.3i)14-s + (−695. − 1.20e3i)15-s + (−825. + 605. i)16-s + (−44.3 + 76.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.809 − 0.586i)2-s + (0.910 − 1.57i)3-s + (0.310 + 0.950i)4-s + (0.438 − 0.759i)5-s + (−1.66 + 0.742i)6-s + 0.0510i·7-s + (0.306 − 0.952i)8-s + (−1.15 − 2.00i)9-s + (−0.800 + 0.357i)10-s − 0.255i·11-s + (1.78 + 0.374i)12-s + (1.17 − 0.681i)13-s + (0.0299 − 0.0413i)14-s + (−0.798 − 1.38i)15-s + (−0.806 + 0.591i)16-s + (−0.0372 + 0.0644i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0413i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0336879 - 1.62806i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0336879 - 1.62806i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.57 + 3.32i)T \) |
| 19 | \( 1 + (1.22e3 - 982. i)T \) |
| good | 3 | \( 1 + (-14.1 + 24.5i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-24.5 + 42.4i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 6.61iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 102. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-718. + 415. i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (44.3 - 76.8i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (-1.77e3 + 1.02e3i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.34e3 - 776. i)T + (1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + 9.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.33e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + (-4.42e3 - 2.55e3i)T + (5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-6.15e3 - 3.55e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.25e4 + 7.26e3i)T + (1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-2.02e4 + 1.17e4i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (8.82e3 - 1.52e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.12e4 + 3.67e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.61e4 + 4.52e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (3.59e4 - 6.21e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-2.86e4 + 4.96e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.91e4 + 5.05e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 5.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + (-9.22e4 + 5.32e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-1.00e5 - 5.78e4i)T + (4.29e9 + 7.43e9i)T^{2} \) |
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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
−13.01558689118144600875493442806, −12.17911944501776756678095844413, −10.78223796872677505303360015673, −9.059196152620234450713895160201, −8.547450387666988663814383504609, −7.51088208646077227114621260803, −6.16072763371350709731510067522, −3.34692266080176252529477866768, −1.85772277762286868541747502411, −0.855041955573625461397454842135,
2.36551601998860542136139815894, 4.08555699622196654267285274443, 5.71395503681480487169708593444, 7.26993126815893489376561935314, 8.855848085235121216416610083951, 9.269348439907878083656425467141, 10.63721097928377424179955288863, 10.92976916518019791240104118006, 13.64150069295023392810678487202, 14.47383859878478382876317353335
