| L(s) = 1 | + (−18.0 − 18.0i)2-s − 19.6i·3-s + 392. i·4-s + (258. − 258. i)5-s + (−354. + 354. i)6-s + 308.·7-s + (2.45e3 − 2.45e3i)8-s + 6.17e3·9-s − 9.31e3·10-s − 2.54e4i·11-s + 7.72e3·12-s + (3.49e4 − 3.49e4i)13-s + (−5.56e3 − 5.56e3i)14-s + (−5.08e3 − 5.08e3i)15-s + 1.19e4·16-s + (−9.79e4 + 9.79e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.12 − 1.12i)2-s − 0.242i·3-s + 1.53i·4-s + (0.413 − 0.413i)5-s + (−0.273 + 0.273i)6-s + 0.128·7-s + (0.600 − 0.600i)8-s + 0.940·9-s − 0.931·10-s − 1.73i·11-s + 0.372·12-s + (1.22 − 1.22i)13-s + (−0.144 − 0.144i)14-s + (−0.100 − 0.100i)15-s + 0.182·16-s + (−1.17 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0399i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.999 - 0.0399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0190960 + 0.955638i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0190960 + 0.955638i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (1.47e6 - 1.15e6i)T \) |
| good | 2 | \( 1 + (18.0 + 18.0i)T + 256iT^{2} \) |
| 3 | \( 1 + 19.6iT - 6.56e3T^{2} \) |
| 5 | \( 1 + (-258. + 258. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 - 308.T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.54e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + (-3.49e4 + 3.49e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (9.79e4 - 9.79e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + (7.13e4 - 7.13e4i)T - 1.69e10iT^{2} \) |
| 23 | \( 1 + (-2.20e5 + 2.20e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (6.49e5 + 6.49e5i)T + 5.00e11iT^{2} \) |
| 31 | \( 1 + (-1.54e5 - 1.54e5i)T + 8.52e11iT^{2} \) |
| 41 | \( 1 + 2.44e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (-3.49e6 + 3.49e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + 3.05e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 3.56e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (6.07e5 - 6.07e5i)T - 1.46e14iT^{2} \) |
| 61 | \( 1 + (-2.06e6 - 2.06e6i)T + 1.91e14iT^{2} \) |
| 67 | \( 1 + 1.64e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.23e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 4.06e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + (4.64e7 - 4.64e7i)T - 1.51e15iT^{2} \) |
| 83 | \( 1 - 8.29e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (2.78e7 + 2.78e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + (3.46e7 - 3.46e7i)T - 7.83e15iT^{2} \) |
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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
−13.38747631575837872742655042995, −12.72834803453516121771523424746, −11.06182510196269653316676042994, −10.48751102487195789804647844430, −8.893680342448129616985790299301, −8.193395829020699563030613004695, −6.01051279179225294342117140412, −3.54521940179116644095720853568, −1.71298692119810767882106287791, −0.59118216863802543892739105925,
1.67130556747384799934339342457, 4.59447648229567139070364810057, 6.62755148220828710707974665461, 7.22752328261506072550674393501, 9.044628825967461811925382714680, 9.724180564610398868942180846894, 11.08091552658495710891883453551, 13.04397027210748098622681797145, 14.53123954409043767453458799132, 15.55126407926019509745108269263
