| L(s) = 1 | + (−5.65 + 9.79i)2-s + (−63.9 − 110. i)4-s + (664. + 383. i)5-s + (1.66e3 + 1.72e3i)7-s + 1.44e3·8-s + (−7.51e3 + 4.33e3i)10-s + (2.52e3 + 4.37e3i)11-s + 2.82e4i·13-s + (−2.63e4 + 6.56e3i)14-s + (−8.19e3 + 1.41e4i)16-s + (−1.35e4 + 7.80e3i)17-s + (1.07e5 + 6.21e4i)19-s − 9.81e4i·20-s − 5.71e4·22-s + (8.97e4 − 1.55e5i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (1.06 + 0.613i)5-s + (0.694 + 0.719i)7-s + 0.353·8-s + (−0.751 + 0.433i)10-s + (0.172 + 0.298i)11-s + 0.990i·13-s + (−0.686 + 0.170i)14-s + (−0.125 + 0.216i)16-s + (−0.161 + 0.0934i)17-s + (0.825 + 0.476i)19-s − 0.613i·20-s − 0.243·22-s + (0.320 − 0.555i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.260855211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260855211\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.65 - 9.79i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.66e3 - 1.72e3i)T \) |
| good | 5 | \( 1 + (-664. - 383. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-2.52e3 - 4.37e3i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 2.82e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (1.35e4 - 7.80e3i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.07e5 - 6.21e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-8.97e4 + 1.55e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 7.11e4T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-8.45e5 + 4.88e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-7.04e5 + 1.21e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 - 3.98e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.21e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-6.69e4 - 3.86e4i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (3.09e6 + 5.36e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.27e7 - 7.37e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (2.14e6 + 1.23e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.79e6 + 3.10e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 3.51e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-3.33e5 + 1.92e5i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.68e7 + 2.91e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 7.92e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (4.51e7 + 2.60e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.44e8iT - 7.83e15T^{2} \) |
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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
−12.08466924583166608267706915977, −11.00785836882559772371753610340, −9.837938640424257566746274574400, −9.107779635633952388092965299470, −7.890657042785724702527011152806, −6.62597296200265862771044235763, −5.81275541065452333469718431817, −4.57340993803665574706112339742, −2.51682258694097683314489184735, −1.43073401975031850784499495914,
0.71881675679159652283891835970, 1.55833164301552373577016964543, 3.04378929817616133936503477353, 4.63418493105832321625857929696, 5.67934617768449494329694197230, 7.32962290591374897635872287088, 8.467510985693009538617291006622, 9.476913668079581331392195838624, 10.37558226954546264851394290958, 11.29501097611212266559463116659
