| L(s) = 1 | + (30.5 + 52.9i)2-s + (−141. + 396. i)3-s + (−844. + 1.46e3i)4-s + (−1.68e3 + 2.91e3i)5-s + (−2.53e4 + 4.60e3i)6-s + (−5.39e3 − 9.35e3i)7-s + 2.19e4·8-s + (−1.36e5 − 1.12e5i)9-s − 2.05e5·10-s + (−1.19e4 − 2.07e4i)11-s + (−4.59e5 − 5.42e5i)12-s + (−1.18e6 + 2.05e6i)13-s + (3.29e5 − 5.71e5i)14-s + (−9.17e5 − 1.08e6i)15-s + (2.40e6 + 4.15e6i)16-s + 8.22e6·17-s + ⋯ |
| L(s) = 1 | + (0.675 + 1.16i)2-s + (−0.336 + 0.941i)3-s + (−0.412 + 0.714i)4-s + (−0.241 + 0.417i)5-s + (−1.32 + 0.241i)6-s + (−0.121 − 0.210i)7-s + 0.236·8-s + (−0.772 − 0.634i)9-s − 0.651·10-s + (−0.0223 − 0.0387i)11-s + (−0.533 − 0.628i)12-s + (−0.885 + 1.53i)13-s + (0.163 − 0.284i)14-s + (−0.311 − 0.367i)15-s + (0.572 + 0.991i)16-s + 1.40·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0107i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.00968889 - 1.80705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00968889 - 1.80705i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (141. - 396. i)T \) |
| good | 2 | \( 1 + (-30.5 - 52.9i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 5 | \( 1 + (1.68e3 - 2.91e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 7 | \( 1 + (5.39e3 + 9.35e3i)T + (-9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (1.19e4 + 2.07e4i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + (1.18e6 - 2.05e6i)T + (-8.96e11 - 1.55e12i)T^{2} \) |
| 17 | \( 1 - 8.22e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 6.96e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + (1.96e7 - 3.39e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + (7.16e7 + 1.24e8i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-7.01e7 + 1.21e8i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 + 9.06e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + (-5.34e8 + 9.24e8i)T + (-2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-5.84e8 - 1.01e9i)T + (-4.64e17 + 8.04e17i)T^{2} \) |
| 47 | \( 1 + (3.32e8 + 5.76e8i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 - 2.63e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-1.28e9 + 2.22e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-1.15e9 - 2.00e9i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (9.01e9 - 1.56e10i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 8.01e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 6.36e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + (1.75e9 + 3.03e9i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + (1.11e10 + 1.93e10i)T + (-6.43e20 + 1.11e21i)T^{2} \) |
| 89 | \( 1 + 2.24e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (5.44e10 + 9.43e10i)T + (-3.57e21 + 6.19e21i)T^{2} \) |
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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
−19.28279554260942377242405901085, −17.10768567988270320492862591800, −16.22822237214287877441995906504, −14.98308646601276843267498783584, −14.00788170741584043482116704633, −11.65522749339383564182537800905, −9.781098032112476395580648891250, −7.33756349597500263677838040460, −5.61502872794652160486524487608, −3.99305839346134321721449594428,
0.889244607096767859624087430435, 2.83151734720338726261389198794, 5.26617717212679473816651570695, 7.78348207230847044292003917318, 10.45178854699402795748449464889, 12.20486849772610442222867440857, 12.62680375990328047557319095002, 14.26052977377822178513651693565, 16.58935175430118275176607703742, 18.24932281473406341628864916600
