| L(s) = 1 | + (−1.17 + 2.03i)2-s + (6.67 + 11.5i)3-s + (13.2 + 22.9i)4-s + (−7.00 − 12.1i)5-s − 31.3·6-s + (52.0 + 90.2i)7-s − 137.·8-s + (32.4 − 56.2i)9-s + 32.9·10-s − 642.·11-s + (−176. + 305. i)12-s + (559. + 969. i)13-s − 245.·14-s + (93.5 − 162. i)15-s + (−261. + 453. i)16-s + (−31.4 + 54.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.207 + 0.360i)2-s + (0.427 + 0.741i)3-s + (0.413 + 0.716i)4-s + (−0.125 − 0.217i)5-s − 0.355·6-s + (0.401 + 0.695i)7-s − 0.759·8-s + (0.133 − 0.231i)9-s + 0.104·10-s − 1.60·11-s + (−0.354 + 0.613i)12-s + (0.918 + 1.59i)13-s − 0.334·14-s + (0.107 − 0.185i)15-s + (−0.255 + 0.442i)16-s + (−0.0264 + 0.0457i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.772859 + 1.43283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.772859 + 1.43283i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (4.78e3 - 6.81e3i)T \) |
| good | 2 | \( 1 + (1.17 - 2.03i)T + (-16 - 27.7i)T^{2} \) |
| 3 | \( 1 + (-6.67 - 11.5i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (7.00 + 12.1i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-52.0 - 90.2i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + 642.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-559. - 969. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (31.4 - 54.5i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (834. + 1.44e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 - 3.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.01e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (-3.72e3 - 6.45e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + 2.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.56e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (5.00e3 - 8.66e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-6.26e3 + 1.08e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-7.42e3 - 1.28e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.21e4 + 3.83e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (3.47e4 + 6.01e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 - 1.61e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-2.65e4 - 4.59e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.18e4 + 5.52e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (4.34e3 - 7.53e3i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.41e5T + 8.58e9T^{2} \) |
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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
−15.66226556711085724434169015731, −15.13990038302508283095557582708, −13.40611097581743189810964365997, −12.08995799787038884913071834371, −10.85854378177466169281408047113, −9.076463776594166989099023583476, −8.349872248930943549239030360185, −6.66495861790797203742939238826, −4.59779801776693722611823705069, −2.77581019613021899894750216026,
1.02582557422197535790344023032, 2.76742162241342604955438988482, 5.45540786036532765574125850815, 7.24576743065780693946763264719, 8.293110751676332224523718796695, 10.46417395180925414961709457639, 10.79927334096807901916604846812, 12.73827250580646433042908917452, 13.59872136477031543116720229836, 14.96554238803187336949493232496
