| L(s) = 1 | + (121.5 + 210. i)3-s + (−2.47e3 + 4.27e3i)5-s + (−1.57e4 + 4.15e4i)7-s + (−2.95e4 + 5.11e4i)9-s + (2.88e5 + 4.99e5i)11-s + 1.93e6·13-s − 1.20e6·15-s + (3.06e6 + 5.30e6i)17-s + (1.10e6 − 1.91e6i)19-s + (−1.06e7 + 1.72e6i)21-s + (−7.04e6 + 1.22e7i)23-s + (1.22e7 + 2.11e7i)25-s − 1.43e7·27-s + 4.59e7·29-s + (−3.58e7 − 6.21e7i)31-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.353 + 0.612i)5-s + (−0.355 + 0.934i)7-s + (−0.166 + 0.288i)9-s + (0.539 + 0.935i)11-s + 1.44·13-s − 0.408·15-s + (0.522 + 0.905i)17-s + (0.102 − 0.177i)19-s + (−0.569 + 0.0923i)21-s + (−0.228 + 0.395i)23-s + (0.249 + 0.432i)25-s − 0.192·27-s + 0.415·29-s + (−0.225 − 0.390i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.299407 + 1.98335i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.299407 + 1.98335i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-121.5 - 210. i)T \) |
| 7 | \( 1 + (1.57e4 - 4.15e4i)T \) |
| good | 5 | \( 1 + (2.47e3 - 4.27e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-2.88e5 - 4.99e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.93e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-3.06e6 - 5.30e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-1.10e6 + 1.91e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (7.04e6 - 1.22e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 4.59e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (3.58e7 + 6.21e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (2.42e8 - 4.19e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 3.28e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 7.73e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-5.37e8 + 9.31e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-9.58e7 - 1.66e8i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.91e9 - 3.32e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-2.37e9 + 4.10e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (3.82e9 + 6.62e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 2.37e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (-2.96e8 - 5.13e8i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (7.44e9 - 1.28e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 3.27e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (3.21e10 - 5.57e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.14e11T + 7.15e21T^{2} \) |
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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
−12.40769884085545384016870707216, −11.39091450315179546505719650062, −10.30059963203429502168000262160, −9.211637694668299578304037582555, −8.226836513820917036551193860289, −6.78540238065909283813671723673, −5.63844597108776776086215459541, −4.02876316867395666152965248195, −3.07368386847987615877838295251, −1.60495377198419842295818932675,
0.53147099546546612790719455722, 1.22472210760582836091051954692, 3.18120358043195163432186650666, 4.17202035435338102547965274811, 5.87777576840251580225315957603, 7.02405272380068177293281062124, 8.231521370954848150581198333337, 9.060656415658690191880103109690, 10.53068176469797844812823572858, 11.61866845576010946937108418572
