| L(s) = 1 | + (0.384 + 0.314i)2-s + (1.96 + 0.158i)3-s + (−0.350 − 1.71i)4-s + (0.992 − 0.120i)5-s + (0.707 + 0.679i)6-s + (2.12 + 3.35i)7-s + (0.866 − 1.65i)8-s + (0.883 + 0.143i)9-s + (0.419 + 0.265i)10-s + (0.846 + 5.20i)11-s + (−0.417 − 3.43i)12-s + (2.97 + 2.04i)13-s + (−0.238 + 1.96i)14-s + (1.97 − 0.0794i)15-s + (−2.37 + 1.01i)16-s + (−3.63 + 2.29i)17-s + ⋯ |
| L(s) = 1 | + (0.272 + 0.222i)2-s + (1.13 + 0.0916i)3-s + (−0.175 − 0.858i)4-s + (0.443 − 0.0539i)5-s + (0.288 + 0.277i)6-s + (0.802 + 1.26i)7-s + (0.306 − 0.583i)8-s + (0.294 + 0.0478i)9-s + (0.132 + 0.0839i)10-s + (0.255 + 1.57i)11-s + (−0.120 − 0.991i)12-s + (0.823 + 0.566i)13-s + (−0.0636 + 0.524i)14-s + (0.509 − 0.0205i)15-s + (−0.593 + 0.252i)16-s + (−0.881 + 0.557i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.81738 + 0.612341i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.81738 + 0.612341i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.992 + 0.120i)T \) |
| 13 | \( 1 + (-2.97 - 2.04i)T \) |
| good | 2 | \( 1 + (-0.384 - 0.314i)T + (0.400 + 1.95i)T^{2} \) |
| 3 | \( 1 + (-1.96 - 0.158i)T + (2.96 + 0.481i)T^{2} \) |
| 7 | \( 1 + (-2.12 - 3.35i)T + (-3.00 + 6.32i)T^{2} \) |
| 11 | \( 1 + (-0.846 - 5.20i)T + (-10.4 + 3.48i)T^{2} \) |
| 17 | \( 1 + (3.63 - 2.29i)T + (7.28 - 15.3i)T^{2} \) |
| 19 | \( 1 + (-0.961 - 0.554i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.85 + 6.67i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.43 + 5.42i)T + (-5.80 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-1.15 + 4.67i)T + (-27.4 - 14.4i)T^{2} \) |
| 37 | \( 1 + (-3.38 + 0.980i)T + (31.2 - 19.7i)T^{2} \) |
| 41 | \( 1 + (-0.605 + 7.50i)T + (-40.4 - 6.57i)T^{2} \) |
| 43 | \( 1 + (1.29 - 4.48i)T + (-36.3 - 22.9i)T^{2} \) |
| 47 | \( 1 + (1.54 - 1.74i)T + (-5.66 - 46.6i)T^{2} \) |
| 53 | \( 1 + (4.56 + 2.39i)T + (30.1 + 43.6i)T^{2} \) |
| 59 | \( 1 + (-2.80 + 6.57i)T + (-40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (-0.296 + 7.35i)T + (-60.8 - 4.90i)T^{2} \) |
| 67 | \( 1 + (8.02 + 1.63i)T + (61.6 + 26.2i)T^{2} \) |
| 71 | \( 1 + (-8.25 + 3.91i)T + (44.9 - 54.9i)T^{2} \) |
| 73 | \( 1 + (-2.33 - 0.883i)T + (54.6 + 48.4i)T^{2} \) |
| 79 | \( 1 + (9.51 + 8.43i)T + (9.52 + 78.4i)T^{2} \) |
| 83 | \( 1 + (-0.928 + 0.640i)T + (29.4 - 77.6i)T^{2} \) |
| 89 | \( 1 + (10.4 - 6.01i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.94 - 11.8i)T + (-26.9 - 93.1i)T^{2} \) |
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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
−9.914067117852885379275567351840, −9.357302733858064278485036819055, −8.659881800307279862075212711460, −7.967199669928419309981610338465, −6.55147688984590146809977535776, −5.96861369771513416617813704038, −4.76112390834585030208305029651, −4.14433526210738044977719427537, −2.26994509484096058835676747753, −1.91285698812009280756841979402,
1.36538417384318315353447772132, 2.90505611480472843291039042605, 3.47939605887726733129605395540, 4.41098739089859375448315591216, 5.63672332206508138039544952065, 6.97026709433444452016781334302, 7.81847513537842483130194031149, 8.472656689130777580472019250954, 8.919631763760937447624540775439, 10.16132197912462114968952128486
