| L(s) = 1 | + (39.1 + 22.6i)2-s + (1.02e3 + 1.77e3i)4-s + (−1.25e4 + 7.23e3i)5-s + (−4.86e4 + 8.42e4i)7-s + 9.26e4i·8-s − 6.54e5·10-s + (−1.20e6 − 6.96e5i)11-s + (3.01e6 + 5.21e6i)13-s + (−3.81e6 + 2.20e6i)14-s + (−2.09e6 + 3.63e6i)16-s − 2.96e7i·17-s − 4.55e7·19-s + (−2.56e7 − 1.48e7i)20-s + (−3.15e7 − 5.46e7i)22-s + (−8.46e7 + 4.88e7i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.801 + 0.462i)5-s + (−0.413 + 0.715i)7-s + 0.353i·8-s − 0.654·10-s + (−0.681 − 0.393i)11-s + (0.623 + 1.08i)13-s + (−0.506 + 0.292i)14-s + (−0.125 + 0.216i)16-s − 1.22i·17-s − 0.967·19-s + (−0.400 − 0.231i)20-s + (−0.278 − 0.481i)22-s + (−0.571 + 0.330i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.7064198888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7064198888\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-39.1 - 22.6i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.25e4 - 7.23e3i)T + (1.22e8 - 2.11e8i)T^{2} \) |
| 7 | \( 1 + (4.86e4 - 8.42e4i)T + (-6.92e9 - 1.19e10i)T^{2} \) |
| 11 | \( 1 + (1.20e6 + 6.96e5i)T + (1.56e12 + 2.71e12i)T^{2} \) |
| 13 | \( 1 + (-3.01e6 - 5.21e6i)T + (-1.16e13 + 2.01e13i)T^{2} \) |
| 17 | \( 1 + 2.96e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 + 4.55e7T + 2.21e15T^{2} \) |
| 23 | \( 1 + (8.46e7 - 4.88e7i)T + (1.09e16 - 1.89e16i)T^{2} \) |
| 29 | \( 1 + (4.17e8 + 2.41e8i)T + (1.76e17 + 3.06e17i)T^{2} \) |
| 31 | \( 1 + (-2.34e8 - 4.06e8i)T + (-3.93e17 + 6.82e17i)T^{2} \) |
| 37 | \( 1 + 4.39e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + (-4.92e9 + 2.84e9i)T + (1.12e19 - 1.95e19i)T^{2} \) |
| 43 | \( 1 + (2.15e9 - 3.72e9i)T + (-1.99e19 - 3.46e19i)T^{2} \) |
| 47 | \( 1 + (3.32e9 + 1.91e9i)T + (5.80e19 + 1.00e20i)T^{2} \) |
| 53 | \( 1 - 2.52e10iT - 4.91e20T^{2} \) |
| 59 | \( 1 + (-5.60e10 + 3.23e10i)T + (8.89e20 - 1.54e21i)T^{2} \) |
| 61 | \( 1 + (-1.61e10 + 2.79e10i)T + (-1.32e21 - 2.29e21i)T^{2} \) |
| 67 | \( 1 + (2.90e10 + 5.03e10i)T + (-4.09e21 + 7.08e21i)T^{2} \) |
| 71 | \( 1 - 3.98e10iT - 1.64e22T^{2} \) |
| 73 | \( 1 - 1.63e11T + 2.29e22T^{2} \) |
| 79 | \( 1 + (-9.35e10 + 1.62e11i)T + (-2.95e22 - 5.11e22i)T^{2} \) |
| 83 | \( 1 + (-6.69e10 - 3.86e10i)T + (5.34e22 + 9.25e22i)T^{2} \) |
| 89 | \( 1 - 9.19e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + (-7.16e11 + 1.24e12i)T + (-3.46e23 - 6.00e23i)T^{2} \) |
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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
−10.86125789126235540402410311977, −9.363779168931519664947201937989, −8.359196474013678492412674743551, −7.29676565620927375741874566510, −6.38735771749444099200290028367, −5.33329316561267564111296466402, −4.08733331701686109719937164485, −3.16550482756318528916415777509, −2.08266574216987689152085675806, −0.14322445744580921308051641097,
0.75405480435737398067853231585, 2.11238673076294519230100454452, 3.56097806217094319545440645465, 4.13922560291901481495103955037, 5.37013702892732815903309693951, 6.50256522663275094873094257307, 7.75995566184847263981977738992, 8.561728978594600957666200930122, 10.23715309013657528048797145124, 10.62958389157109264445725485841
