L(s) = 1 | + (3.54 − 6.13i)2-s + (−9.08 − 15.7i)4-s + (−39.9 + 69.1i)5-s + (43.1 + 122. i)7-s + 97.9·8-s + (282. + 489. i)10-s + (175. + 304. i)11-s − 291.·13-s + (902. + 168. i)14-s + (637. − 1.10e3i)16-s + (−185. − 320. i)17-s + (−752. + 1.30e3i)19-s + 1.45e3·20-s + 2.49e3·22-s + (−212. + 368. i)23-s + ⋯ |
L(s) = 1 | + (0.626 − 1.08i)2-s + (−0.283 − 0.491i)4-s + (−0.713 + 1.23i)5-s + (0.332 + 0.942i)7-s + 0.541·8-s + (0.893 + 1.54i)10-s + (0.438 + 0.759i)11-s − 0.478·13-s + (1.23 + 0.229i)14-s + (0.622 − 1.07i)16-s + (−0.155 − 0.268i)17-s + (−0.478 + 0.828i)19-s + 0.810·20-s + 1.09·22-s + (−0.0839 + 0.145i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.391i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.919 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.08878 + 0.426420i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.08878 + 0.426420i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-43.1 - 122. i)T \) |
good | 2 | \( 1 + (-3.54 + 6.13i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (39.9 - 69.1i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-175. - 304. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 291.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (185. + 320. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (752. - 1.30e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (212. - 368. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 7.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-1.28e3 - 2.23e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (369. - 640. i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 7.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.83e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (766. - 1.32e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (4.76e3 + 8.25e3i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.48e4 + 2.56e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.32e4 + 4.02e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.33e4 + 2.31e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 1.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.50e4 - 6.07e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-1.35e4 + 2.34e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 7.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-2.17e4 + 3.77e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−14.09809795969035264658099080488, −12.44624390453632393874494440888, −11.90189921264168712488343183786, −10.95223612730783218675917488434, −9.893283831515083083584705392544, −8.040780793052691957782564082760, −6.71346355715129733091805075076, −4.73458097908058953024985863570, −3.31265362931352781397155158593, −2.11763828385506265396121672891,
0.867141472878998590617865335301, 4.14113630618112886663513500779, 4.94295543733906248290774680853, 6.53172781641714239469902512070, 7.78351928278602256665776457931, 8.711884023178081049163715470371, 10.55273099568533605101876896911, 11.89183494484657478897741506402, 13.14834119740066171576691001842, 13.95172127208168416479548769333