| L(s) = 1 | + (366. + 634. i)3-s + (−4.49e3 + 7.77e3i)5-s + (4.30e4 + 1.12e4i)7-s + (−1.79e5 + 3.11e5i)9-s + (4.56e4 + 7.91e4i)11-s − 7.75e5·13-s − 6.57e6·15-s + (−4.63e6 − 8.03e6i)17-s + (−1.43e6 + 2.48e6i)19-s + (8.60e6 + 3.14e7i)21-s + (5.26e6 − 9.12e6i)23-s + (−1.59e7 − 2.75e7i)25-s − 1.33e8·27-s − 4.24e7·29-s + (−5.29e7 − 9.16e7i)31-s + ⋯ |
| L(s) = 1 | + (0.869 + 1.50i)3-s + (−0.642 + 1.11i)5-s + (0.967 + 0.253i)7-s + (−1.01 + 1.75i)9-s + (0.0855 + 0.148i)11-s − 0.579·13-s − 2.23·15-s + (−0.792 − 1.37i)17-s + (−0.133 + 0.230i)19-s + (0.459 + 1.67i)21-s + (0.170 − 0.295i)23-s + (−0.326 − 0.564i)25-s − 1.78·27-s − 0.384·29-s + (−0.331 − 0.574i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.108599600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108599600\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-4.30e4 - 1.12e4i)T \) |
| good | 3 | \( 1 + (-366. - 634. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 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 | \( 1 + 1.39e11T + 7.15e21T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.71986352298522992705280318141, −11.05044174506571724860753429480, −10.12410689385277583293442754516, −9.123119633675420579243920404339, −8.103497353348503576779982223389, −7.06567839183609240216624975251, −5.12142296794916684714976535478, −4.26894670091775809230455478011, −3.14829603450817577949477307894, −2.26884791869804920471970618224,
0.20973567701931250925891467655, 1.32590706168622647074129003752, 2.06151600859470409566010353788, 3.71801222847508383587977698130, 5.00143401075411045989527311899, 6.61879258393210721690496484364, 7.74905907285607665257933539665, 8.312257054313243050180884788414, 9.082992905852206227718641981105, 11.00937340596207368061964893401
