L(s) = 1 | + (−1.73 + i)2-s + (1.5 + 0.866i)3-s + (0.999 − 1.73i)4-s + (−0.866 − 0.5i)5-s − 3.46·6-s + (−1.73 + i)7-s + (1.5 + 2.59i)9-s + 1.99·10-s + (5.19 − 3i)11-s + (2.99 − 1.73i)12-s + (3.23 − 1.59i)13-s + (1.99 − 3.46i)14-s + (−0.866 − 1.5i)15-s + (1.99 + 3.46i)16-s − 17-s + (−5.19 − 3i)18-s + ⋯ |
L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.866 + 0.499i)3-s + (0.499 − 0.866i)4-s + (−0.387 − 0.223i)5-s − 1.41·6-s + (−0.654 + 0.377i)7-s + (0.5 + 0.866i)9-s + 0.632·10-s + (1.56 − 0.904i)11-s + (0.866 − 0.500i)12-s + (0.896 − 0.443i)13-s + (0.534 − 0.925i)14-s + (−0.223 − 0.387i)15-s + (0.499 + 0.866i)16-s − 0.242·17-s + (−1.22 − 0.707i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.664421 + 0.741941i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.664421 + 0.741941i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (-3.23 + 1.59i)T \) |
good | 2 | \( 1 + (1.73 - i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.19 + 3i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.92 - 4i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + (-1.73 - i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (8.66 - 5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + (-5.19 - 3i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-8.66 + 5i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.44850348129599342823979492884, −9.758130987141156448104424648202, −8.917788227400866796410289927992, −8.498938245570724661852554332540, −7.79301199580912705483573447502, −6.58438856348517467870276561944, −5.86129372743986573456806269248, −3.99733983962288186047700822684, −3.37431005155124049813486343894, −1.31908878548689850259178556920,
0.923451805834034125913335803134, 2.16296743276808025450618966523, 3.34617826599026920783086675271, 4.36964616414684454371753312904, 6.66431488351843470087323458027, 6.91133765759704696590064958901, 8.138730555944762309582064154136, 8.912190065310129832997154333236, 9.465300012981883236625980594521, 10.16485128636909388205433896882