| L(s) = 1 | − 32i·2-s − 1.02e3·4-s + (6.89e3 + 1.13e3i)5-s + 1.43e4i·7-s + 3.27e4i·8-s + (3.61e4 − 2.20e5i)10-s − 4.93e5·11-s + 9.84e5i·13-s + 4.60e5·14-s + 1.04e6·16-s − 9.98e6i·17-s − 9.97e6·19-s + (−7.06e6 − 1.15e6i)20-s + 1.57e7i·22-s − 4.66e6i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.986 + 0.161i)5-s + 0.323i·7-s + 0.353i·8-s + (0.114 − 0.697i)10-s − 0.924·11-s + 0.735i·13-s + 0.228·14-s + 0.250·16-s − 1.70i·17-s − 0.924·19-s + (−0.493 − 0.0809i)20-s + 0.653i·22-s − 0.151i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.25727 - 1.48028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25727 - 1.48028i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 32iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-6.89e3 - 1.13e3i)T \) |
| good | 7 | \( 1 - 1.43e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 4.93e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 9.84e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 9.98e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 + 9.97e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.66e6iT - 9.52e14T^{2} \) |
| 29 | \( 1 - 4.31e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.06e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.33e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 1.24e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.57e9iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 3.78e8iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 3.54e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 - 6.43e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.83e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 7.62e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 1.48e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.24e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 3.51e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 9.64e9iT - 1.28e21T^{2} \) |
| 89 | \( 1 + 4.39e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.45e11iT - 7.15e21T^{2} \) |
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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
−11.53455984189990590559403540733, −10.47125834085807561288012015912, −9.577793067475085036262388020421, −8.619270825368912569312628718070, −7.01925475487646675448341681876, −5.65722476915691396939638913985, −4.58455754007430294314562917774, −2.80490144745316059449076687581, −2.07405233524716523092090156845, −0.52721811993465941155334658503,
1.03338546796581251215225176377, 2.54282306844757839709199420596, 4.25619370824149681684371435158, 5.56216304177598894962226264041, 6.34087867185546868512651144323, 7.78719538501883087825259578074, 8.707740085406296191918176302430, 10.06515511039191896586307084213, 10.70746519799647033105471947194, 12.73216288008401875307322643888
