| L(s) = 1 | + (2.29 − 1.92i)3-s + (−2.38 − 4.12i)5-s + (3.51 + 6.09i)7-s + (1.55 − 8.86i)9-s + (2.56 + 4.44i)11-s − 18.0i·13-s + (−13.4 − 4.88i)15-s + (−0.508 + 0.881i)17-s + (−0.399 − 18.9i)19-s + (19.8 + 7.21i)21-s − 23.4·23-s + (1.14 − 1.98i)25-s + (−13.5 − 23.3i)27-s + (39.6 + 22.9i)29-s + (−2.83 − 1.63i)31-s + ⋯ |
| L(s) = 1 | + (0.765 − 0.643i)3-s + (−0.476 − 0.825i)5-s + (0.502 + 0.870i)7-s + (0.172 − 0.984i)9-s + (0.233 + 0.403i)11-s − 1.38i·13-s + (−0.895 − 0.325i)15-s + (−0.0299 + 0.0518i)17-s + (−0.0210 − 0.999i)19-s + (0.944 + 0.343i)21-s − 1.01·23-s + (0.0458 − 0.0793i)25-s + (−0.500 − 0.865i)27-s + (1.36 + 0.789i)29-s + (−0.0914 − 0.0528i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.047791882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.047791882\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.29 + 1.92i)T \) |
| 19 | \( 1 + (0.399 + 18.9i)T \) |
| good | 5 | \( 1 + (2.38 + 4.12i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-3.51 - 6.09i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-2.56 - 4.44i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 18.0iT - 169T^{2} \) |
| 17 | \( 1 + (0.508 - 0.881i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + 23.4T + 529T^{2} \) |
| 29 | \( 1 + (-39.6 - 22.9i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (2.83 + 1.63i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 57.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-48.8 + 28.2i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + 79.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (33.1 - 57.3i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (36.1 - 20.8i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-14.3 + 8.31i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-39.3 + 68.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + 120. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (40.9 + 23.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (47.5 - 82.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 - 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-13.8 - 23.9i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-24.4 + 14.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 101. iT - 9.40e3T^{2} \) |
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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
−9.754809656800170775342303324164, −8.874408859591380779487230493924, −8.292605114957056405559111510477, −7.68764414907239131857625623521, −6.54708983624596101618990644753, −5.41312391154609709701206043190, −4.45737070908540595015095853380, −3.15119226021965722737194944893, −2.05217357938856923306021044806, −0.67467290354671984347190346798,
1.69768534134367026046508812355, 3.10634904445249805667957190513, 4.00566678467832713955537983689, 4.65960659913646308663698129329, 6.27093235847861823601315080014, 7.18195500493884119829905046339, 8.029792156889613017470275614821, 8.690945581097551900447799752868, 9.984066639655664777123256054170, 10.27086473289209045772067160501
