| L(s) = 1 | + (−4.82 − 10.0i)2-s + (11.1 + 16.3i)3-s + (82.3 − 103. i)4-s + (−812. + 876. i)5-s + (110. − 191. i)6-s + (876. − 506. i)7-s + (−4.21e3 − 961. i)8-s + (2.25e3 − 5.74e3i)9-s + (1.27e4 + 3.92e3i)10-s + (9.87e3 + 1.23e4i)11-s + (2.61e3 + 195. i)12-s + (8.50e3 − 2.62e3i)13-s + (−9.31e3 − 6.34e3i)14-s + (−2.34e4 − 3.53e3i)15-s + (3.17e3 + 1.39e4i)16-s + (8.98e4 − 8.34e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.301 − 0.626i)2-s + (0.137 + 0.202i)3-s + (0.321 − 0.403i)4-s + (−1.30 + 1.40i)5-s + (0.0852 − 0.147i)6-s + (0.365 − 0.210i)7-s + (−1.02 − 0.234i)8-s + (0.343 − 0.875i)9-s + (1.27 + 0.392i)10-s + (0.674 + 0.845i)11-s + (0.126 + 0.00944i)12-s + (0.297 − 0.0918i)13-s + (−0.242 − 0.165i)14-s + (−0.463 − 0.0698i)15-s + (0.0484 + 0.212i)16-s + (1.07 − 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.25489 - 0.812991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25489 - 0.812991i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.87e6 - 2.85e6i)T \) |
| good | 2 | \( 1 + (4.82 + 10.0i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (-11.1 - 16.3i)T + (-2.39e3 + 6.10e3i)T^{2} \) |
| 5 | \( 1 + (812. - 876. i)T + (-2.91e4 - 3.89e5i)T^{2} \) |
| 7 | \( 1 + (-876. + 506. i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-9.87e3 - 1.23e4i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (-8.50e3 + 2.62e3i)T + (6.73e8 - 4.59e8i)T^{2} \) |
| 17 | \( 1 + (-8.98e4 + 8.34e4i)T + (5.21e8 - 6.95e9i)T^{2} \) |
| 19 | \( 1 + (7.11e4 - 2.79e4i)T + (1.24e10 - 1.15e10i)T^{2} \) |
| 23 | \( 1 + (-4.19e5 + 6.32e4i)T + (7.48e10 - 2.30e10i)T^{2} \) |
| 29 | \( 1 + (-2.28e5 + 3.34e5i)T + (-1.82e11 - 4.65e11i)T^{2} \) |
| 31 | \( 1 + (-1.00e5 + 1.34e6i)T + (-8.43e11 - 1.27e11i)T^{2} \) |
| 37 | \( 1 + (1.17e6 + 6.77e5i)T + (1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-2.42e6 + 1.16e6i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (1.66e6 - 2.08e6i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-7.01e6 - 2.16e6i)T + (5.14e13 + 3.50e13i)T^{2} \) |
| 59 | \( 1 + (-2.22e6 - 9.76e6i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-2.26e7 + 1.69e6i)T + (1.89e14 - 2.85e13i)T^{2} \) |
| 67 | \( 1 + (8.42e6 + 2.14e7i)T + (-2.97e14 + 2.76e14i)T^{2} \) |
| 71 | \( 1 + (9.25e5 - 6.13e6i)T + (-6.17e14 - 1.90e14i)T^{2} \) |
| 73 | \( 1 + (2.71e6 + 8.81e6i)T + (-6.66e14 + 4.54e14i)T^{2} \) |
| 79 | \( 1 + (1.09e7 + 1.90e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-3.95e7 + 2.69e7i)T + (8.22e14 - 2.09e15i)T^{2} \) |
| 89 | \( 1 + (-2.22e7 - 3.26e7i)T + (-1.43e15 + 3.66e15i)T^{2} \) |
| 97 | \( 1 + (-3.79e7 - 4.76e7i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−14.62152007009431169892303085172, −12.23055077827522258554676517871, −11.47112255927476700719226634709, −10.53154092180184755534819416151, −9.410762429324263429624424499768, −7.50620249755099169486055781847, −6.52714118504946445642335492157, −4.02501503064422538464823524521, −2.81085484549664563493896090868, −0.77483766727562488761526132167,
1.16295948177919740543155494156, 3.61663479118920368474894335318, 5.22044507833247413370742645141, 7.12445817315260765819538983189, 8.395694469162808071092134554634, 8.617515944274367183422482841347, 11.11245018499193998741187882865, 12.14182014123350776528457355269, 13.00813884426356706438034606834, 14.75177162764015600452158115178
