| L(s) = 1 | + (1.99 + 0.156i)2-s + (−1.43 + 2.63i)3-s + (3.95 + 0.624i)4-s + (0.898 + 5.09i)5-s + (−3.28 + 5.02i)6-s + (0.391 + 0.328i)7-s + (7.77 + 1.86i)8-s + (−4.85 − 7.57i)9-s + (0.993 + 10.2i)10-s + (0.754 − 4.27i)11-s + (−7.32 + 9.50i)12-s + (−3.92 + 10.7i)13-s + (0.728 + 0.716i)14-s + (−14.7 − 4.96i)15-s + (15.2 + 4.93i)16-s + (−19.7 + 11.3i)17-s + ⋯ |
| L(s) = 1 | + (0.996 + 0.0783i)2-s + (−0.479 + 0.877i)3-s + (0.987 + 0.156i)4-s + (0.179 + 1.01i)5-s + (−0.546 + 0.837i)6-s + (0.0559 + 0.0469i)7-s + (0.972 + 0.233i)8-s + (−0.539 − 0.841i)9-s + (0.0993 + 1.02i)10-s + (0.0685 − 0.389i)11-s + (−0.610 + 0.791i)12-s + (−0.301 + 0.828i)13-s + (0.0520 + 0.0511i)14-s + (−0.980 − 0.331i)15-s + (0.951 + 0.308i)16-s + (−1.15 + 0.669i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61522 + 1.83228i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61522 + 1.83228i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.99 - 0.156i)T \) |
| 3 | \( 1 + (1.43 - 2.63i)T \) |
| good | 5 | \( 1 + (-0.898 - 5.09i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-0.391 - 0.328i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-0.754 + 4.27i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (3.92 - 10.7i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (19.7 - 11.3i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-12.0 - 6.94i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (9.73 + 11.6i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-28.6 + 10.4i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-47.1 + 39.5i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (14.8 - 8.58i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (12.1 - 33.3i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (5.52 + 0.974i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-30.0 + 35.7i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 29.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-4.81 - 27.2i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-26.7 + 31.8i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-29.5 + 81.2i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-31.3 + 18.1i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (0.520 - 0.900i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-51.5 + 18.7i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (131. - 47.9i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-80.8 - 46.7i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (6.08 - 34.5i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.09456239160597989827871974281, −11.45367690528497936276678429503, −10.62754920875544410652408967365, −9.836048147656865474146668999752, −8.296587351108409156486764352265, −6.66341732755691414969022833804, −6.22542582534705433700345014322, −4.82751741094753265587834029015, −3.83090094050846327000509680299, −2.53817792866028929924515094485,
1.14456734811110962990668987751, 2.68864217139321059947867245361, 4.69584006957916283065490766158, 5.31991460167091563572512959613, 6.56009046154840226191773475014, 7.48030384242406295471074614042, 8.663703854477372075307903447066, 10.16453575382012058084542645068, 11.26947546793870902300479960287, 12.18506416968006733300420166745
