| L(s) = 1 | + (−121.5 − 210. i)3-s + (84.6 − 146. i)5-s + (−9.85e3 − 4.33e4i)7-s + (−2.95e4 + 5.11e4i)9-s + (−1.43e5 − 2.49e5i)11-s + 2.48e6·13-s − 4.11e4·15-s + (−3.58e6 − 6.20e6i)17-s + (−1.63e6 + 2.83e6i)19-s + (−7.92e6 + 7.34e6i)21-s + (1.56e7 − 2.71e7i)23-s + (2.43e7 + 4.22e7i)25-s + 1.43e7·27-s + 5.63e7·29-s + (−8.04e7 − 1.39e8i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.0121 − 0.0209i)5-s + (−0.221 − 0.975i)7-s + (−0.166 + 0.288i)9-s + (−0.269 − 0.466i)11-s + 1.85·13-s − 0.0139·15-s + (−0.611 − 1.05i)17-s + (−0.151 + 0.262i)19-s + (−0.423 + 0.392i)21-s + (0.507 − 0.878i)23-s + (0.499 + 0.865i)25-s + 0.192·27-s + 0.509·29-s + (−0.504 − 0.874i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.182717 - 1.26552i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.182717 - 1.26552i\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (9.85e3 + 4.33e4i)T \) |
| good | 5 | \( 1 + (-84.6 + 146. i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (1.43e5 + 2.49e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 2.48e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + (3.58e6 + 6.20e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (1.63e6 - 2.83e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-1.56e7 + 2.71e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 5.63e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + (8.04e7 + 1.39e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-1.48e8 + 2.57e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 1.39e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 4.76e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.19e9 - 2.06e9i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (2.34e9 + 4.05e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (8.21e8 + 1.42e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (1.55e9 - 2.69e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (5.33e9 + 9.24e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 2.59e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (2.45e8 + 4.24e8i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (4.61e9 - 7.98e9i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 3.48e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.81e10 + 3.15e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.51e11T + 7.15e21T^{2} \) |
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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
−11.23002338251911740040133985313, −10.83224649576370272538958124466, −9.250980618729899668984537920179, −8.054497395088422744126200010962, −6.88915776823460191593149804307, −5.92211525431090732162944694880, −4.37300307047264004050817224852, −3.03610094901021772123873702702, −1.29541620885019854436623603688, −0.35539552801845018368993985220,
1.43548957345395872865165536733, 2.98708502487632949616497844701, 4.29999818486148717915082807876, 5.66378175448727865800421789166, 6.54527926736133611943746535234, 8.362041920261080823774238349011, 9.126990206636730083649453434451, 10.46997114386926621020249598323, 11.29959863382975691849337620879, 12.49780052105323003396688586845
