| L(s) = 1 | + (−2.85 + 4.94i)2-s + (1.00e3 + 1.74e3i)4-s + (4.49e3 − 7.77e3i)5-s + (−4.30e4 − 1.12e4i)7-s − 2.31e4·8-s + (2.56e4 + 4.44e4i)10-s + (4.56e4 + 7.91e4i)11-s − 7.75e5·13-s + (1.78e5 − 1.80e5i)14-s + (−1.99e6 + 3.45e6i)16-s + (4.63e6 + 8.03e6i)17-s + (1.43e6 − 2.48e6i)19-s + 1.81e7·20-s − 5.21e5·22-s + (5.26e6 − 9.12e6i)23-s + ⋯ |
| L(s) = 1 | + (−0.0630 + 0.109i)2-s + (0.492 + 0.852i)4-s + (0.642 − 1.11i)5-s + (−0.967 − 0.253i)7-s − 0.250·8-s + (0.0810 + 0.140i)10-s + (0.0855 + 0.148i)11-s − 0.579·13-s + (0.0887 − 0.0896i)14-s + (−0.476 + 0.824i)16-s + (0.792 + 1.37i)17-s + (0.133 − 0.230i)19-s + 1.26·20-s − 0.0215·22-s + (0.170 − 0.295i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.43935 + 1.18533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43935 + 1.18533i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (4.30e4 + 1.12e4i)T \) |
| good | 2 | \( 1 + (2.85 - 4.94i)T + (-1.02e3 - 1.77e3i)T^{2} \) |
| 5 | \( 1 + (-4.49e3 + 7.77e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-4.56e4 - 7.91e4i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 7.75e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-4.63e6 - 8.03e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-1.43e6 + 2.48e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-5.26e6 + 9.12e6i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 4.24e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-5.29e7 - 9.16e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (1.54e8 - 2.68e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 1.19e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.16e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.25e8 - 2.16e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.73e9 - 3.00e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-3.01e9 - 5.22e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (5.31e9 - 9.21e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (1.10e9 + 1.91e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.34e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-8.02e9 - 1.38e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (8.41e9 - 1.45e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 1.25e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (3.02e10 - 5.23e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.39e11T + 7.15e21T^{2} \) |
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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
−12.64617833900065155850684692980, −12.24505730934041376451892352068, −10.47559096364416267272639435185, −9.329744034695541417291349150270, −8.281662620156525240697488448090, −6.95712328401387857205105882554, −5.74524128526645267399738577451, −4.14696767354599333816559067443, −2.74079888232071934090325441956, −1.19593055460043651357809497516,
0.52950325104032784882343182952, 2.25247404827293084556573553278, 3.11872708129693155292890512867, 5.39119055183948346234890029657, 6.38738874336388258420125563818, 7.26676909613303469366626256890, 9.481698331617924400186820094646, 9.991360620296425757706297674986, 11.08632763006305772517131899846, 12.23354597271664141205715610511
