| L(s) = 1 | + (1.89 + 0.629i)2-s + (−1.96 − 2.26i)3-s + (3.20 + 2.39i)4-s + (5.84 − 3.37i)5-s + (−2.31 − 5.53i)6-s + (−3.50 − 2.02i)7-s + (4.58 + 6.55i)8-s + (−1.24 + 8.91i)9-s + (13.2 − 2.72i)10-s + (−4.13 + 7.16i)11-s + (−0.904 − 11.9i)12-s + (7.66 − 4.42i)13-s + (−5.38 − 6.05i)14-s + (−19.1 − 6.58i)15-s + (4.57 + 15.3i)16-s − 28.6·17-s + ⋯ |
| L(s) = 1 | + (0.949 + 0.314i)2-s + (−0.656 − 0.754i)3-s + (0.801 + 0.597i)4-s + (1.16 − 0.675i)5-s + (−0.385 − 0.922i)6-s + (−0.501 − 0.289i)7-s + (0.572 + 0.819i)8-s + (−0.138 + 0.990i)9-s + (1.32 − 0.272i)10-s + (−0.376 + 0.651i)11-s + (−0.0753 − 0.997i)12-s + (0.589 − 0.340i)13-s + (−0.384 − 0.432i)14-s + (−1.27 − 0.439i)15-s + (0.285 + 0.958i)16-s − 1.68·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.86188 - 0.212230i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86188 - 0.212230i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.89 - 0.629i)T \) |
| 3 | \( 1 + (1.96 + 2.26i)T \) |
| good | 5 | \( 1 + (-5.84 + 3.37i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (3.50 + 2.02i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (4.13 - 7.16i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-7.66 + 4.42i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 28.6T + 289T^{2} \) |
| 19 | \( 1 + 7.93T + 361T^{2} \) |
| 23 | \( 1 + (25.3 - 14.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-15.7 - 9.08i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-40.8 + 23.5i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 13.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-31.0 - 53.7i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-26.5 + 46.0i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (12.8 + 7.43i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 100. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (11.4 + 19.9i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (51.3 + 29.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 17.9i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 54.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-62.9 - 36.3i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-4.34 + 7.53i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 28.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (0.200 - 0.348i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.80623143665642475513526072437, −13.25738894564060954061457730799, −12.64094516324498162801724022069, −11.34580361903369924728059066828, −10.05688004343647067744870054146, −8.254254248881134945841320216706, −6.72161794291710784648794512042, −5.92200372401865411302492381585, −4.66263592344820795116014325263, −2.10111781607429616860403854723,
2.66374704220442627412016687771, 4.40225779292735302379730433110, 6.07799870865417563182144397468, 6.35594452051347371849889122262, 9.163036637498921461640059088337, 10.41038667516218654758350067137, 10.92731304977852939465697127184, 12.24705821767181182695064752620, 13.49275599249299224785610619088, 14.22784474124651672788552418973
