| L(s) = 1 | + (1.05 − 0.224i)2-s + (−0.766 + 0.341i)4-s + (1.13 + 1.92i)5-s + (0.316 + 0.548i)7-s + (−2.47 + 1.79i)8-s + (1.63 + 1.77i)10-s + (−5.10 + 1.08i)11-s + (3.72 + 0.791i)13-s + (0.456 + 0.506i)14-s + (−1.08 + 1.20i)16-s + (0.365 − 0.265i)17-s + (−3.97 + 2.88i)19-s + (−1.52 − 1.08i)20-s + (−5.13 + 2.28i)22-s + (0.0686 + 0.0762i)23-s + ⋯ |
| L(s) = 1 | + (0.745 − 0.158i)2-s + (−0.383 + 0.170i)4-s + (0.509 + 0.860i)5-s + (0.119 + 0.207i)7-s + (−0.874 + 0.635i)8-s + (0.515 + 0.560i)10-s + (−1.53 + 0.327i)11-s + (1.03 + 0.219i)13-s + (0.121 + 0.135i)14-s + (−0.270 + 0.300i)16-s + (0.0886 − 0.0643i)17-s + (−0.911 + 0.662i)19-s + (−0.341 − 0.242i)20-s + (−1.09 + 0.487i)22-s + (0.0143 + 0.0158i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00987 + 1.22482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00987 + 1.22482i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.13 - 1.92i)T \) |
| good | 2 | \( 1 + (-1.05 + 0.224i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (-0.316 - 0.548i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.10 - 1.08i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.72 - 0.791i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 0.265i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (3.97 - 2.88i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.0686 - 0.0762i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (1.04 - 9.98i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (0.422 + 4.01i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.0800 - 0.246i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-7.65 - 1.62i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-3.68 - 6.38i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.10 + 10.5i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-4.56 - 3.31i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-7.00 - 1.48i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (4.50 - 0.958i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (0.364 + 3.46i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-3.47 - 2.52i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.75 - 14.6i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.142 - 1.35i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (1.95 + 0.870i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-5.42 + 16.7i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.637 + 6.06i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−10.77642554743380526378452182233, −10.08624368380607152915684066819, −8.968580251417674930131914863423, −8.185655016823447426035187023980, −7.14102140514143572644943662620, −5.94355498205379609700632364762, −5.39511573142524400145213706525, −4.19042430443081824292005677759, −3.15570105736834334938866230307, −2.18589506582708185133053876630,
0.65921155590403170201376771799, 2.51883421316283149543565646149, 3.96091168470239336037512375065, 4.78926988186420575141890758976, 5.66978784632238941086057952103, 6.21699017806711590563672073899, 7.74880737413025770296423233236, 8.583956556457905227246899148548, 9.301173982208982469400235889625, 10.31490018102302416162936718900
