L(s) = 1 | + (−45.0 − 4.65i)2-s + (13.6 − 13.6i)3-s + (2.00e3 + 419. i)4-s + (−3.50e3 − 3.50e3i)5-s + (−675. + 548. i)6-s + 3.76e4i·7-s + (−8.82e4 − 2.82e4i)8-s + 1.76e5i·9-s + (1.41e5 + 1.73e5i)10-s + (−2.83e5 − 2.83e5i)11-s + (3.29e4 − 2.15e4i)12-s + (1.70e6 − 1.70e6i)13-s + (1.75e5 − 1.69e6i)14-s − 9.52e4·15-s + (3.84e6 + 1.68e6i)16-s + 1.57e6·17-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.102i)2-s + (0.0323 − 0.0323i)3-s + (0.978 + 0.204i)4-s + (−0.501 − 0.501i)5-s + (−0.0354 + 0.0288i)6-s + 0.846i·7-s + (−0.952 − 0.304i)8-s + 0.997i·9-s + (0.446 + 0.550i)10-s + (−0.530 − 0.530i)11-s + (0.0382 − 0.0250i)12-s + (1.27 − 1.27i)13-s + (0.0871 − 0.842i)14-s − 0.0323·15-s + (0.916 + 0.400i)16-s + 0.269·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.926301 - 0.394528i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.926301 - 0.394528i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (45.0 + 4.65i)T \) |
good | 3 | \( 1 + (-13.6 + 13.6i)T - 1.77e5iT^{2} \) |
| 5 | \( 1 + (3.50e3 + 3.50e3i)T + 4.88e7iT^{2} \) |
| 7 | \( 1 - 3.76e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (2.83e5 + 2.83e5i)T + 2.85e11iT^{2} \) |
| 13 | \( 1 + (-1.70e6 + 1.70e6i)T - 1.79e12iT^{2} \) |
| 17 | \( 1 - 1.57e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (-1.25e7 + 1.25e7i)T - 1.16e14iT^{2} \) |
| 23 | \( 1 - 6.83e6iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (-5.00e7 + 5.00e7i)T - 1.22e16iT^{2} \) |
| 31 | \( 1 - 7.19e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-9.81e7 - 9.81e7i)T + 1.77e17iT^{2} \) |
| 41 | \( 1 + 7.24e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (-1.93e8 - 1.93e8i)T + 9.29e17iT^{2} \) |
| 47 | \( 1 - 1.39e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (3.32e9 + 3.32e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (-5.46e9 - 5.46e9i)T + 3.01e19iT^{2} \) |
| 61 | \( 1 + (4.88e9 - 4.88e9i)T - 4.35e19iT^{2} \) |
| 67 | \( 1 + (7.47e9 - 7.47e9i)T - 1.22e20iT^{2} \) |
| 71 | \( 1 - 2.58e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 1.88e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 3.09e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + (-2.96e10 + 2.96e10i)T - 1.28e21iT^{2} \) |
| 89 | \( 1 + 6.42e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 8.87e10T + 7.15e21T^{2} \) |
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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
−16.05851319805227435329534394544, −15.64014768332296791717011625036, −13.28638817172062028974867766299, −11.76361298591856454115068467781, −10.53758261881596327212212039920, −8.726898236443444932842504665281, −7.83405715596612267775207153541, −5.60199133102614284451515429221, −2.81700371115127833264682695327, −0.74698789235937475480991847399,
1.16918475730803376399271951682, 3.55731008385668612819583218704, 6.48811404801039366944486019299, 7.72055924344267004138613431128, 9.410802810640770008749337777307, 10.77148656944959433888466635013, 12.00492132369150519462939398719, 14.18168335724811328675742240067, 15.54652941130837600732087621568, 16.62110543262935987678668949945