| L(s) = 1 | + (56.4 − 56.4i)3-s + (6.26e3 + 6.26e3i)5-s + 3.29e4i·7-s + 1.70e5i·9-s + (1.51e5 + 1.51e5i)11-s + (−1.05e6 + 1.05e6i)13-s + 7.06e5·15-s − 1.95e6·17-s + (−5.25e6 + 5.25e6i)19-s + (1.86e6 + 1.86e6i)21-s − 4.62e7i·23-s + 2.95e7i·25-s + (1.96e7 + 1.96e7i)27-s + (1.16e8 − 1.16e8i)29-s − 2.86e8·31-s + ⋯ |
| L(s) = 1 | + (0.134 − 0.134i)3-s + (0.895 + 0.895i)5-s + 0.740i·7-s + 0.964i·9-s + (0.284 + 0.284i)11-s + (−0.785 + 0.785i)13-s + 0.240·15-s − 0.334·17-s + (−0.486 + 0.486i)19-s + (0.0993 + 0.0993i)21-s − 1.49i·23-s + 0.605i·25-s + (0.263 + 0.263i)27-s + (1.05 − 1.05i)29-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.412420 + 1.62531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.412420 + 1.62531i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (-56.4 + 56.4i)T - 1.77e5iT^{2} \) |
| 5 | \( 1 + (-6.26e3 - 6.26e3i)T + 4.88e7iT^{2} \) |
| 7 | \( 1 - 3.29e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (-1.51e5 - 1.51e5i)T + 2.85e11iT^{2} \) |
| 13 | \( 1 + (1.05e6 - 1.05e6i)T - 1.79e12iT^{2} \) |
| 17 | \( 1 + 1.95e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (5.25e6 - 5.25e6i)T - 1.16e14iT^{2} \) |
| 23 | \( 1 + 4.62e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (-1.16e8 + 1.16e8i)T - 1.22e16iT^{2} \) |
| 31 | \( 1 + 2.86e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-3.33e8 - 3.33e8i)T + 1.77e17iT^{2} \) |
| 41 | \( 1 - 1.31e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (5.60e8 + 5.60e8i)T + 9.29e17iT^{2} \) |
| 47 | \( 1 + 1.10e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + (2.81e9 + 2.81e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (-3.57e8 - 3.57e8i)T + 3.01e19iT^{2} \) |
| 61 | \( 1 + (6.15e9 - 6.15e9i)T - 4.35e19iT^{2} \) |
| 67 | \( 1 + (8.27e9 - 8.27e9i)T - 1.22e20iT^{2} \) |
| 71 | \( 1 - 1.51e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 1.11e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 1.68e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + (-1.04e10 + 1.04e10i)T - 1.28e21iT^{2} \) |
| 89 | \( 1 - 3.00e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 1.53e11T + 7.15e21T^{2} \) |
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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
−13.16756924975336089267591047400, −11.96141869339573084151229756975, −10.68327623896683747361758251848, −9.764958511026653312235377441248, −8.495428367051489843228915414730, −7.02864332018580502970783538605, −6.02155373785156765504401545794, −4.59227101961496039051579723477, −2.56030679771753471135389084382, −1.97823134678454044732769898632,
0.40831668546380029804614516795, 1.55253186882019270949280815643, 3.32333616930947172034755888175, 4.76961138343016073100731352594, 5.98207407758677482517732713353, 7.37738514504047892279621648507, 8.949722826027152351951476284316, 9.620543022015530924266983110288, 10.87464031358994825684611098857, 12.38073975819104072008736532029
