| L(s) = 1 | + (−1.39 − 0.245i)2-s + (2.66 − 1.37i)3-s + (1.87 + 0.684i)4-s + (−1.49 − 1.77i)5-s + (−4.05 + 1.26i)6-s + (5.91 − 2.15i)7-s + (−2.44 − 1.41i)8-s + (5.20 − 7.33i)9-s + (1.64 + 2.84i)10-s + (1.00 − 1.20i)11-s + (5.95 − 0.764i)12-s + (1.33 + 7.56i)13-s + (−8.76 + 1.54i)14-s + (−6.43 − 2.68i)15-s + (3.06 + 2.57i)16-s + (−20.1 + 11.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 − 0.122i)2-s + (0.888 − 0.458i)3-s + (0.469 + 0.171i)4-s + (−0.298 − 0.355i)5-s + (−0.675 + 0.210i)6-s + (0.844 − 0.307i)7-s + (−0.306 − 0.176i)8-s + (0.578 − 0.815i)9-s + (0.164 + 0.284i)10-s + (0.0916 − 0.109i)11-s + (0.495 − 0.0637i)12-s + (0.102 + 0.581i)13-s + (−0.625 + 0.110i)14-s + (−0.428 − 0.179i)15-s + (0.191 + 0.160i)16-s + (−1.18 + 0.684i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.03780 - 0.391894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03780 - 0.391894i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.245i)T \) |
| 3 | \( 1 + (-2.66 + 1.37i)T \) |
| good | 5 | \( 1 + (1.49 + 1.77i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-5.91 + 2.15i)T + (37.5 - 31.4i)T^{2} \) |
| 11 | \( 1 + (-1.00 + 1.20i)T + (-21.0 - 119. i)T^{2} \) |
| 13 | \( 1 + (-1.33 - 7.56i)T + (-158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (20.1 - 11.6i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (15.0 - 26.1i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (7.69 - 21.1i)T + (-405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-49.0 - 8.65i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (27.8 + 10.1i)T + (736. + 617. i)T^{2} \) |
| 37 | \( 1 + (-14.5 - 25.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-17.1 + 3.02i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (64.1 + 53.7i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-11.3 - 31.1i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + 86.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-29.2 - 34.9i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-79.4 + 28.9i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (8.80 + 49.9i)T + (-4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (32.5 - 18.7i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (3.93 - 6.82i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (12.5 - 71.1i)T + (-5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (25.4 + 4.47i)T + (6.47e3 + 2.35e3i)T^{2} \) |
| 89 | \( 1 + (23.4 + 13.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (27.0 + 22.6i)T + (1.63e3 + 9.26e3i)T^{2} \) |
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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
−14.94089866328298589184029510749, −13.95865412544738781321315496257, −12.66076444745489473456762280086, −11.52931635215837254157840032737, −10.13960870618966698778545021678, −8.647260881240783858309510930375, −8.086325948341846036816438161047, −6.63600865927231771640720021908, −4.07723058900772414390851245447, −1.78401267701721940044924095631,
2.55374762041958283778299267696, 4.68845264259736452791407064947, 6.95406418364898416851889688276, 8.254977594345423952659443509269, 9.064119107212924483221265574706, 10.50824199642014831231108996765, 11.39773471923444296542219603078, 13.15192355988045919847708771907, 14.52793772444276584532682942997, 15.24512312976223084000493504955
