| L(s) = 1 | + (53.9 − 31.1i)2-s + (401. + 695. i)3-s + (918. − 1.59e3i)4-s + 673. i·5-s + (4.33e4 + 2.50e4i)6-s + (−9.26e3 − 5.34e3i)7-s + 1.31e4i·8-s + (−2.34e5 + 4.05e5i)9-s + (2.09e4 + 3.63e4i)10-s + (4.51e5 − 2.60e5i)11-s + 1.47e6·12-s + (4.51e5 − 1.26e6i)13-s − 6.66e5·14-s + (−4.68e5 + 2.70e5i)15-s + (2.29e6 + 3.96e6i)16-s + (3.89e6 − 6.74e6i)17-s + ⋯ |
| L(s) = 1 | + (1.19 − 0.688i)2-s + (0.954 + 1.65i)3-s + (0.448 − 0.776i)4-s + 0.0963i·5-s + (2.27 + 1.31i)6-s + (−0.208 − 0.120i)7-s + 0.142i·8-s + (−1.32 + 2.28i)9-s + (0.0663 + 0.114i)10-s + (0.845 − 0.488i)11-s + 1.71·12-s + (0.337 − 0.941i)13-s − 0.331·14-s + (−0.159 + 0.0919i)15-s + (0.546 + 0.946i)16-s + (0.665 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.83384 + 1.37800i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.83384 + 1.37800i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-4.51e5 + 1.26e6i)T \) |
| good | 2 | \( 1 + (-53.9 + 31.1i)T + (1.02e3 - 1.77e3i)T^{2} \) |
| 3 | \( 1 + (-401. - 695. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 - 673. iT - 4.88e7T^{2} \) |
| 7 | \( 1 + (9.26e3 + 5.34e3i)T + (9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-4.51e5 + 2.60e5i)T + (1.42e11 - 2.47e11i)T^{2} \) |
| 17 | \( 1 + (-3.89e6 + 6.74e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (7.00e6 + 4.04e6i)T + (5.82e13 + 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-2.44e6 - 4.23e6i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (9.48e7 + 1.64e8i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 - 1.03e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 + (4.72e8 - 2.72e8i)T + (8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + (2.83e8 - 1.63e8i)T + (2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-9.73e6 + 1.68e7i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + 3.29e8iT - 2.47e18T^{2} \) |
| 53 | \( 1 - 2.39e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (-3.95e9 - 2.28e9i)T + (1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-2.03e9 + 3.52e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (1.64e10 - 9.47e9i)T + (6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + (-1.35e10 - 7.81e9i)T + (1.15e20 + 2.00e20i)T^{2} \) |
| 73 | \( 1 - 7.66e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 1.53e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.49e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + (6.73e10 - 3.88e10i)T + (1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + (-7.31e10 - 4.22e10i)T + (3.57e21 + 6.19e21i)T^{2} \) |
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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
−16.79668669312579986655564063958, −15.39603598999287130349400653761, −14.41835077647954646425427852751, −13.42589194057809696703224100874, −11.41445114004027499949438962145, −10.16260672140604737204051638103, −8.609636539451421873500937768935, −5.22638308523898857135662243018, −3.82372264691164449006226038883, −2.86636579938346639592063201656,
1.58039969288293526826243040705, 3.64387592035209661148969396631, 6.21385975710361302443736053917, 7.18650163123085146045239271027, 8.872387456713529939957044052498, 12.24769808242313055363652755456, 12.94936071419381757670723010238, 14.24392697031314222277865157792, 14.82442609928040706675923247920, 16.86589489688086637979037807393
