L(s) = 1 | + (3.30 − 5.72i)3-s + (12.6 − 21.8i)5-s + 40.7·7-s + (99.6 + 172. i)9-s − 324.·11-s + (−48.8 − 84.6i)13-s + (−83.3 − 144. i)15-s + (−1.05e3 + 1.83e3i)17-s + (−806. − 1.35e3i)19-s + (134. − 232. i)21-s + (−1.50e3 − 2.60e3i)23-s + (1.24e3 + 2.15e3i)25-s + 2.92e3·27-s + (−527. − 914. i)29-s − 7.60e3·31-s + ⋯ |
L(s) = 1 | + (0.211 − 0.367i)3-s + (0.225 − 0.390i)5-s + 0.314·7-s + (0.410 + 0.710i)9-s − 0.809·11-s + (−0.0801 − 0.138i)13-s + (−0.0955 − 0.165i)15-s + (−0.887 + 1.53i)17-s + (−0.512 − 0.858i)19-s + (0.0665 − 0.115i)21-s + (−0.593 − 1.02i)23-s + (0.398 + 0.689i)25-s + 0.771·27-s + (−0.116 − 0.201i)29-s − 1.42·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5459910534\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5459910534\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (806. + 1.35e3i)T \) |
good | 3 | \( 1 + (-3.30 + 5.72i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-12.6 + 21.8i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 40.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 324.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (48.8 + 84.6i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (1.05e3 - 1.83e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (1.50e3 + 2.60e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (527. + 914. i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 7.60e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.05e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-230. + 399. i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-188. + 326. i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-8.35e3 - 1.44e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-5.23e3 - 9.06e3i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (2.37e4 - 4.11e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.37e4 + 2.37e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.72e4 + 2.98e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (1.87e3 - 3.25e3i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-6.14e3 + 1.06e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.67e4 - 4.62e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 1.31e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-5.17e4 - 8.96e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (5.29e4 - 9.17e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−10.86345833014588451928621495022, −10.61615276810527350796463491388, −9.168804916320467813267687056031, −8.322779403182795683058712841172, −7.52630359526646026603258994123, −6.36853782785318513886931237162, −5.16399172813402949256055720550, −4.22849443468866907331006503240, −2.49729266772358759987882702129, −1.56842241896837183568690314595,
0.12914676380206150994374138560, 1.89986052989185577436389145030, 3.15731780410240033048917161364, 4.34137369372075044436773359066, 5.45925839062771723477391165566, 6.69093183525171171948391439336, 7.58107125671497592038813242341, 8.777074562658561731815976152465, 9.647083691877555458890722064163, 10.43790445842628978337996812257