L(s) = 1 | + (−1.23 + 0.688i)2-s + (1.05 − 1.70i)4-s + (0.317 − 0.550i)5-s + (−2.11 − 1.58i)7-s + (−0.130 + 2.82i)8-s + (−0.0138 + 0.898i)10-s + (−3.16 + 1.82i)11-s − 4.15i·13-s + (3.70 + 0.499i)14-s + (−1.78 − 3.58i)16-s + (−3.01 + 1.74i)17-s + (−1.99 + 3.45i)19-s + (−0.601 − 1.11i)20-s + (2.65 − 4.43i)22-s + (−1.47 + 2.54i)23-s + ⋯ |
L(s) = 1 | + (−0.873 + 0.486i)2-s + (0.526 − 0.850i)4-s + (0.142 − 0.246i)5-s + (−0.800 − 0.598i)7-s + (−0.0462 + 0.998i)8-s + (−0.00438 + 0.284i)10-s + (−0.954 + 0.550i)11-s − 1.15i·13-s + (0.991 + 0.133i)14-s + (−0.445 − 0.895i)16-s + (−0.732 + 0.422i)17-s + (−0.457 + 0.791i)19-s + (−0.134 − 0.250i)20-s + (0.565 − 0.945i)22-s + (−0.306 + 0.530i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0110117 - 0.0660475i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0110117 - 0.0660475i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.23 - 0.688i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.11 + 1.58i)T \) |
good | 5 | \( 1 + (-0.317 + 0.550i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.16 - 1.82i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4.15iT - 13T^{2} \) |
| 17 | \( 1 + (3.01 - 1.74i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.99 - 3.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.47 - 2.54i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.35T + 29T^{2} \) |
| 31 | \( 1 + (5.20 - 3.00i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.59 + 0.923i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.4iT - 41T^{2} \) |
| 43 | \( 1 + 2.83T + 43T^{2} \) |
| 47 | \( 1 + (-4.61 + 7.99i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.99 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.10 - 5.25i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.72 + 0.995i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.01 + 13.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 0.737T + 71T^{2} \) |
| 73 | \( 1 + (-2.13 - 3.70i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.74 - 4.47i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.67iT - 83T^{2} \) |
| 89 | \( 1 + (-4.63 - 2.67i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−10.51098760857672452063987484091, −9.615583059752560735829666455363, −8.763244074643724093901578329424, −7.71543189044062368187700584698, −7.12803770816216082849025184798, −5.94725559715065137468497260825, −5.16351703774208054436018602899, −3.52445988145546198144257394342, −1.93664889177545359069846011303, −0.04826603174219196780261847962,
2.19943607495682901414156165809, 3.03128267505279656336477580648, 4.46688440598413526293406885977, 6.09126262356345527953524672448, 6.81074664925799271621339518388, 7.88087267392723291332906216059, 8.987299852759073628828298046686, 9.348757226680496850735896830915, 10.47784817335439337603920840497, 11.13560890142412809598710807088