L(s) = 1 | + (10.5 − 11.4i)3-s − 101. i·5-s − 13.9i·7-s + (−18.9 − 242. i)9-s − 445.·11-s + 217.·13-s + (−1.16e3 − 1.07e3i)15-s + 1.43e3i·17-s − 2.61e3i·19-s + (−159. − 147. i)21-s − 696.·23-s − 7.26e3·25-s + (−2.97e3 − 2.34e3i)27-s − 4.60e3i·29-s + 8.33e3i·31-s + ⋯ |
L(s) = 1 | + (0.679 − 0.734i)3-s − 1.82i·5-s − 0.107i·7-s + (−0.0778 − 0.996i)9-s − 1.11·11-s + 0.356·13-s + (−1.33 − 1.23i)15-s + 1.20i·17-s − 1.65i·19-s + (−0.0789 − 0.0730i)21-s − 0.274·23-s − 2.32·25-s + (−0.784 − 0.619i)27-s − 1.01i·29-s + 1.55i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.679i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.444133755\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.444133755\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-10.5 + 11.4i)T \) |
good | 5 | \( 1 + 101. iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 13.9iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 445.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 217.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.43e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.61e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 696.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.60e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.33e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.16e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 6.05e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.78e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.11e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 203.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.28e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.85e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 7.61e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.52e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 8.78e4T + 8.58e9T^{2} \) |
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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
−9.654841369186260131887122380376, −8.734742220441454165333230285596, −8.319637658433491971457575643707, −7.37963813306724487427459548763, −6.04771474902009150525538937411, −5.01701803132040979999446430132, −3.96406227070619688367702004979, −2.44713519363347119374417755344, −1.27938982775671147536617523203, −0.31931483484182881792786362875,
2.20744328732978592326559843117, 2.99570677233485009012964224041, 3.84098906185728560203819870330, 5.28890664038313554018140750785, 6.37935071130650014859454397854, 7.59212666326358514398848216890, 8.081232278825760383293915376562, 9.608726433626470894744137756858, 10.11160716138129136532970425949, 10.91988456163244923218097413352