| L(s) = 1 | + (−1.93 − 1.62i)2-s + (0.613 − 0.223i)3-s + (0.766 + 4.34i)4-s + (−0.233 + 1.32i)5-s + (−1.55 − 0.565i)6-s + (−0.766 − 1.32i)7-s + (3.05 − 5.28i)8-s + (−1.97 + 1.65i)9-s + (2.61 − 2.19i)10-s + (0.592 − 1.02i)11-s + (1.43 + 2.49i)12-s + (−2.55 − 0.929i)13-s + (−0.673 + 3.82i)14-s + (0.152 + 0.866i)15-s + (−6.23 + 2.27i)16-s + (2.97 + 2.49i)17-s + ⋯ |
| L(s) = 1 | + (−1.37 − 1.15i)2-s + (0.354 − 0.128i)3-s + (0.383 + 2.17i)4-s + (−0.104 + 0.593i)5-s + (−0.634 − 0.230i)6-s + (−0.289 − 0.501i)7-s + (1.07 − 1.86i)8-s + (−0.657 + 0.551i)9-s + (0.826 − 0.693i)10-s + (0.178 − 0.309i)11-s + (0.415 + 0.719i)12-s + (−0.708 − 0.257i)13-s + (−0.180 + 1.02i)14-s + (0.0394 + 0.223i)15-s + (−1.55 + 0.567i)16-s + (0.720 + 0.604i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.315586 - 0.182066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.315586 - 0.182066i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-0.819 + 4.28i)T \) |
| good | 2 | \( 1 + (1.93 + 1.62i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.613 + 0.223i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (0.233 - 1.32i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.766 + 1.32i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.592 + 1.02i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.55 + 0.929i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.97 - 2.49i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.879 + 4.98i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.56 - 2.99i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.91 - 3.32i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.10T + 37T^{2} \) |
| 41 | \( 1 + (-9.38 + 3.41i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.51 - 8.57i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.439 + 0.368i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.511 - 2.89i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (3.01 + 2.52i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.784 + 4.44i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.97 - 2.49i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.20 - 6.83i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (5.75 - 2.09i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (9.21 - 3.35i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (6.15 + 10.6i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.27 - 0.829i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.64 - 4.73i)T + (16.8 + 95.5i)T^{2} \) |
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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
−18.77735995923535259005713061412, −17.46286449521392914520972112470, −16.50519445136723328310599643142, −14.41031108850262875564062938710, −12.72471086044817483942803291073, −11.20109023371604474172051081790, −10.26542560881370856685877618346, −8.755142049334680624551706331763, −7.39209480653299014951293223245, −2.95600261473892036517366115513,
5.78031012960520657860516432631, 7.60518620624050872929393087320, 8.964246542321939847993066642946, 9.789451133347758811425143701092, 12.01330198531743448583992967123, 14.31303028238350861606711702864, 15.34010531092120958757460543924, 16.49171658055940940778840062233, 17.38903320656237636774339173691, 18.64769786126024164034748744806
