L(s) = 1 | + (1.02 − 0.593i)2-s + (−0.295 + 0.511i)4-s + (−1.44 − 1.71i)5-s + (1.75 + 1.01i)7-s + 3.07i·8-s + (−2.49 − 0.903i)10-s + (1.94 + 3.36i)11-s + (2.96 − 2.05i)13-s + 2.40·14-s + (1.23 + 2.14i)16-s + (4.71 + 2.72i)17-s + (2.94 − 5.09i)19-s + (1.29 − 0.231i)20-s + (3.99 + 2.30i)22-s + (0.298 − 0.172i)23-s + ⋯ |
L(s) = 1 | + (0.727 − 0.419i)2-s + (−0.147 + 0.255i)4-s + (−0.644 − 0.764i)5-s + (0.664 + 0.383i)7-s + 1.08i·8-s + (−0.789 − 0.285i)10-s + (0.585 + 1.01i)11-s + (0.821 − 0.570i)13-s + 0.644·14-s + (0.308 + 0.535i)16-s + (1.14 + 0.660i)17-s + (0.674 − 1.16i)19-s + (0.290 − 0.0517i)20-s + (0.850 + 0.491i)22-s + (0.0623 − 0.0359i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.04656 + 0.00405593i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.04656 + 0.00405593i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.44 + 1.71i)T \) |
| 13 | \( 1 + (-2.96 + 2.05i)T \) |
good | 2 | \( 1 + (-1.02 + 0.593i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-1.75 - 1.01i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.94 - 3.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-4.71 - 2.72i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.94 + 5.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.298 + 0.172i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 + (4.71 - 2.72i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0902 - 0.156i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.15 - 0.669i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12.2iT - 47T^{2} \) |
| 53 | \( 1 + 2.42iT - 53T^{2} \) |
| 59 | \( 1 + (-3.53 + 6.11i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.38 - 5.85i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.81 + 2.20i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.940 - 1.62i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 8.86iT - 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 7.83iT - 83T^{2} \) |
| 89 | \( 1 + (6.12 + 10.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.02 + 2.90i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.17485718344582205816837130320, −9.811024864724412956261453149504, −8.770050502722248583680405631684, −8.159301280572262365774120257877, −7.32535182407559275110890460956, −5.72628537177131329586449444854, −4.88965431517569525955397979652, −4.13114624911555259815072975768, −3.10549081509625716194294002069, −1.49442307084098923167413859699,
1.17564206107977052159223287520, 3.43708033704031881580614468253, 3.92166248092144946925227075403, 5.22800658792741331395201005386, 6.07417504724232302029895156370, 6.99144641161313004787603311853, 7.82400449408494050136796938844, 8.853344290099002411758020508332, 9.979967345737376578879884945087, 10.79081588868704006650901053921