| L(s) = 1 | + (−0.326 − 0.896i)2-s + (−0.266 − 0.0971i)3-s + (0.834 − 0.700i)4-s + 0.270i·6-s + (−0.0138 − 0.0783i)7-s + (−2.55 − 1.47i)8-s + (−2.23 − 1.87i)9-s + (2.24 − 3.87i)11-s + (−0.290 + 0.105i)12-s + (−0.539 − 0.642i)13-s + (−0.0657 + 0.0379i)14-s + (−0.109 + 0.622i)16-s + (−3.04 + 3.63i)17-s + (−0.952 + 2.61i)18-s + (−0.698 + 1.92i)19-s + ⋯ |
| L(s) = 1 | + (−0.230 − 0.633i)2-s + (−0.154 − 0.0560i)3-s + (0.417 − 0.350i)4-s + 0.110i·6-s + (−0.00522 − 0.0296i)7-s + (−0.902 − 0.521i)8-s + (−0.745 − 0.625i)9-s + (0.675 − 1.16i)11-s + (−0.0839 + 0.0305i)12-s + (−0.149 − 0.178i)13-s + (−0.0175 + 0.0101i)14-s + (−0.0274 + 0.155i)16-s + (−0.739 + 0.881i)17-s + (−0.224 + 0.616i)18-s + (−0.160 + 0.440i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0188982 + 0.919596i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0188982 + 0.919596i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 37 | \( 1 + (4.52 + 4.06i)T \) |
| good | 2 | \( 1 + (0.326 + 0.896i)T + (-1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (0.266 + 0.0971i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (0.0138 + 0.0783i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.24 + 3.87i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.539 + 0.642i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.04 - 3.63i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (0.698 - 1.92i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.00393 + 0.00227i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.499 - 0.288i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.06iT - 31T^{2} \) |
| 41 | \( 1 + (4.55 - 3.81i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + 1.01iT - 43T^{2} \) |
| 47 | \( 1 + (3.97 + 6.88i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.138 - 0.788i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (7.84 + 1.38i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.86 + 3.40i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.563 - 3.19i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.63 - 2.05i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 - 9.38T + 73T^{2} \) |
| 79 | \( 1 + (-6.85 + 1.20i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (10.7 + 9.01i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (10.5 + 1.86i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (6.13 - 3.54i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.731513988987854238577096271266, −8.904646803217502653932039594469, −8.278279329435166942109173990383, −6.84029864450765505795581885499, −6.17987600697494169209289661189, −5.53982048806365260100581437043, −3.89654721165521811479859481671, −3.09644312713989372077336026111, −1.82310686179166067775917382838, −0.44227181180569610039370075206,
2.04373162220135567465604657764, 3.03586159460019913093905089575, 4.50182481304824358810298173329, 5.33591136691372450346055742652, 6.52969759953329393344423134553, 7.00595966230988962782023901593, 7.88431564579683451541826821397, 8.779623092797330568672534674228, 9.397965383216287790230200343918, 10.55032432929849511763166661950
