| L(s) = 1 | + (−0.118 − 1.57i)2-s + (−3.07 − 0.948i)3-s + (−0.494 + 0.0745i)4-s + (0.291 − 0.270i)5-s + (−1.13 + 4.95i)6-s + (2.43 − 1.02i)7-s + (−0.527 − 2.31i)8-s + (6.07 + 4.13i)9-s + (−0.460 − 0.427i)10-s + (−1.83 + 1.25i)11-s + (1.59 + 0.239i)12-s + (1.92 + 0.928i)13-s + (−1.91 − 3.72i)14-s + (−1.15 + 0.554i)15-s + (−4.53 + 1.40i)16-s + (0.243 + 0.620i)17-s + ⋯ |
| L(s) = 1 | + (−0.0835 − 1.11i)2-s + (−1.77 − 0.547i)3-s + (−0.247 + 0.0372i)4-s + (0.130 − 0.120i)5-s + (−0.462 + 2.02i)6-s + (0.921 − 0.389i)7-s + (−0.186 − 0.817i)8-s + (2.02 + 1.37i)9-s + (−0.145 − 0.135i)10-s + (−0.552 + 0.376i)11-s + (0.459 + 0.0692i)12-s + (0.534 + 0.257i)13-s + (−0.510 − 0.994i)14-s + (−0.297 + 0.143i)15-s + (−1.13 + 0.350i)16-s + (0.0591 + 0.150i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.279742 - 0.504088i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.279742 - 0.504088i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.43 + 1.02i)T \) |
| good | 2 | \( 1 + (0.118 + 1.57i)T + (-1.97 + 0.298i)T^{2} \) |
| 3 | \( 1 + (3.07 + 0.948i)T + (2.47 + 1.68i)T^{2} \) |
| 5 | \( 1 + (-0.291 + 0.270i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (1.83 - 1.25i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-1.92 - 0.928i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-0.243 - 0.620i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-0.781 - 1.35i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.231 - 0.589i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-3.83 + 4.80i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.69 + 2.93i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.20 + 0.633i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-1.37 - 6.02i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (1.37 - 6.03i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-0.409 - 5.46i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (9.05 - 1.36i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-6.10 - 5.66i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (9.38 + 1.41i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-0.892 + 1.54i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.30 + 1.63i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-1.08 + 14.4i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-0.821 - 1.42i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.20 - 2.50i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (10.5 + 7.21i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + 9.59T + 97T^{2} \) |
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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
−15.55450614284309809988555258310, −13.50356717146781904658988844059, −12.53057607621136278979236672440, −11.55929861441951156726230009923, −10.96677898361607711077411503378, −9.977263499794098461958706483764, −7.57989056003280225675615176590, −6.15689785930600094817229045227, −4.62328488574438392718599804750, −1.47151453892123708233529014291,
4.94092103452733077691792280745, 5.75249499872341029866115541025, 6.91418324028772708905288756961, 8.508986226207788847390973048155, 10.44017148695969488222123615094, 11.28114912036435463244013577599, 12.26802060335634561536354111476, 14.13835604766318598268132017942, 15.54571369548249491876110464155, 15.93374455873561168224008228348
