| L(s) = 1 | + (−26.5 + 5.03i)3-s + (−212. − 122. i)5-s + (−198. − 343. i)7-s + (678. − 267. i)9-s + (426. − 246. i)11-s + (−1.43e3 + 2.48e3i)13-s + (6.24e3 + 2.17e3i)15-s − 928. i·17-s − 1.00e4·19-s + (6.99e3 + 8.12e3i)21-s + (−1.33e3 − 771. i)23-s + (2.21e4 + 3.83e4i)25-s + (−1.66e4 + 1.05e4i)27-s + (2.92e4 − 1.68e4i)29-s + (7.23e3 − 1.25e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.982 + 0.186i)3-s + (−1.69 − 0.979i)5-s + (−0.578 − 1.00i)7-s + (0.930 − 0.366i)9-s + (0.320 − 0.185i)11-s + (−0.653 + 1.13i)13-s + (1.84 + 0.645i)15-s − 0.189i·17-s − 1.46·19-s + (0.755 + 0.876i)21-s + (−0.109 − 0.0633i)23-s + (1.41 + 2.45i)25-s + (−0.845 + 0.533i)27-s + (1.19 − 0.692i)29-s + (0.242 − 0.420i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.243168 + 0.144693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.243168 + 0.144693i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (26.5 - 5.03i)T \) |
| good | 5 | \( 1 + (212. + 122. i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (198. + 343. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-426. + 246. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.43e3 - 2.48e3i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + 928. iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.00e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.33e3 + 771. i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.92e4 + 1.68e4i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-7.23e3 + 1.25e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 3.16e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-2.79e4 - 1.61e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (1.75e4 + 3.04e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-6.58e4 + 3.79e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 - 8.98e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (1.81e5 + 1.04e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.09e5 - 1.89e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.45e5 - 4.25e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 6.95e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.74e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (4.61e4 + 7.98e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-2.80e5 + 1.61e5i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 - 7.41e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-5.17e5 - 8.96e5i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−13.28171601792022767533690347566, −12.21646556391211219019345634247, −11.61406026196628987477997050773, −10.45237101869473319893194990548, −9.040713092094881726631470184962, −7.60376923907286962936448277626, −6.54972053083331831839067654605, −4.53453281262809658737890035996, −4.06076962262125960218755218313, −0.78181557373701484384306137600,
0.19459521606091837969351253438, 2.88507607197143916018022386003, 4.45544065352999328673474261561, 6.14576734331485746085860127772, 7.16053457708443585186553931849, 8.339844182881033295689614963491, 10.24030320292665355771611783489, 11.10817102245850674031327733121, 12.21893455896283048852633350665, 12.59326453374136933122909051578
