L(s) = 1 | + (1.37 + 2.38i)2-s + (−2.80 + 4.85i)4-s + (0.5 + 0.866i)5-s − 2.84·7-s − 9.94·8-s + (−1.37 + 2.38i)10-s + 0.864·11-s + (−0.321 + 0.557i)13-s + (−3.92 − 6.80i)14-s + (−8.11 − 14.0i)16-s + (1.87 + 3.24i)17-s + (−3.36 − 2.77i)19-s − 5.60·20-s + (1.19 + 2.06i)22-s + (0.208 − 0.361i)23-s + ⋯ |
L(s) = 1 | + (0.975 + 1.68i)2-s + (−1.40 + 2.42i)4-s + (0.223 + 0.387i)5-s − 1.07·7-s − 3.51·8-s + (−0.436 + 0.755i)10-s + 0.260·11-s + (−0.0892 + 0.154i)13-s + (−1.04 − 1.81i)14-s + (−2.02 − 3.51i)16-s + (0.453 + 0.785i)17-s + (−0.770 − 0.636i)19-s − 1.25·20-s + (0.254 + 0.440i)22-s + (0.0435 − 0.0753i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.835259 - 1.23256i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.835259 - 1.23256i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (3.36 + 2.77i)T \) |
good | 2 | \( 1 + (-1.37 - 2.38i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + 2.84T + 7T^{2} \) |
| 11 | \( 1 - 0.864T + 11T^{2} \) |
| 13 | \( 1 + (0.321 - 0.557i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.87 - 3.24i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.208 + 0.361i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.85 - 8.40i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4.93T + 31T^{2} \) |
| 37 | \( 1 - 6.36T + 37T^{2} \) |
| 41 | \( 1 + (2.00 + 3.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.02 - 1.78i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.97 - 3.42i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.49 - 9.51i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.22 - 2.13i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.16 - 5.48i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.26 + 2.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.891 + 1.54i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.56 - 6.17i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.912 + 1.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.43T + 83T^{2} \) |
| 89 | \( 1 + (-2.22 + 3.85i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.42 - 9.39i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.72505704250787658259090750828, −9.515365227349482141921561761963, −8.886583705809772140148935348959, −7.896955081016784288208372947880, −7.01838934902261350171524738534, −6.39656746448314994907894469208, −5.81842266812332649424729689534, −4.69823984405448521196899132880, −3.73920699839670109988785380090, −2.87918855079306428133464893516,
0.51066131467419042088468104233, 1.99218221040240746535129833470, 3.03879938354027395747823524729, 3.90083547441996642315630084675, 4.81963481344565008355690764325, 5.82365582275884945487578511500, 6.48272029808297644737798805935, 8.196715362965055267513701145512, 9.544169734691492201627223314167, 9.605534692094072041527012983893