| L(s) = 1 | + (0.0147 − 0.00259i)2-s + (−1.87 + 0.683i)4-s + (−1.54 + 0.562i)5-s + (2.21 − 1.45i)7-s + (−0.0517 + 0.0298i)8-s + (−0.0212 + 0.0122i)10-s + (−1.57 + 4.31i)11-s + (−1.52 − 4.19i)13-s + (0.0287 − 0.0271i)14-s + (3.06 − 2.57i)16-s + (−1.88 − 3.26i)17-s + (−1.27 − 0.734i)19-s + (2.52 − 2.11i)20-s + (−0.0119 + 0.0675i)22-s + (−7.31 − 1.29i)23-s + ⋯ |
| L(s) = 1 | + (0.0103 − 0.00183i)2-s + (−0.939 + 0.341i)4-s + (−0.691 + 0.251i)5-s + (0.835 − 0.549i)7-s + (−0.0182 + 0.0105i)8-s + (−0.00672 + 0.00388i)10-s + (−0.473 + 1.30i)11-s + (−0.423 − 1.16i)13-s + (0.00767 − 0.00724i)14-s + (0.765 − 0.642i)16-s + (−0.457 − 0.792i)17-s + (−0.291 − 0.168i)19-s + (0.563 − 0.473i)20-s + (−0.00253 + 0.0143i)22-s + (−1.52 − 0.269i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.151840 - 0.327939i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151840 - 0.327939i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.21 + 1.45i)T \) |
| good | 2 | \( 1 + (-0.0147 + 0.00259i)T + (1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (1.54 - 0.562i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (1.57 - 4.31i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.52 + 4.19i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.88 + 3.26i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.27 + 0.734i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.31 + 1.29i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.83 + 7.78i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.87 + 7.89i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + 0.794T + 37T^{2} \) |
| 41 | \( 1 + (3.97 - 1.44i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.303 + 1.71i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.54 - 0.563i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (3.66 + 2.11i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.85 - 7.43i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.02 + 2.82i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.88 - 10.6i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (6.66 + 3.84i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 15.8iT - 73T^{2} \) |
| 79 | \( 1 + (-0.276 - 1.56i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (4.61 + 1.68i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.59 + 6.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.56 - 0.276i)T + (91.1 - 33.1i)T^{2} \) |
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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
−10.18602801327829066516431923131, −9.791894179274305857328130251287, −8.394602390825434390331706057669, −7.77768607119048859016408592341, −7.25033692164115303152867094852, −5.55360462203684193195646405540, −4.54092637916524848902265322716, −3.98847669075575053616972529305, −2.42318680372020656434537247763, −0.20845304028262548063389998353,
1.73155386057183925775221029265, 3.58913446201870759163507483988, 4.53355191455960545635947009440, 5.36769745911530526215285775877, 6.37204524175739410084912163853, 7.86233633555241797687114355437, 8.540223989009440188396412742056, 8.983401340705787417372275036413, 10.27002759140763232139095162848, 11.03077778667407433050122455557
