L(s) = 1 | + (−0.347 + 1.96i)4-s + (−0.173 − 0.984i)7-s + (3.83 + 3.21i)13-s + (−3.75 − 1.36i)16-s + (3.5 + 6.06i)19-s + (−3.83 + 3.21i)25-s + 1.99·28-s + (−0.694 + 3.93i)31-s + (−5.5 + 9.52i)37-s + (−7.51 − 2.73i)43-s + (5.63 − 2.05i)49-s + (−7.66 + 6.42i)52-s + (−0.173 − 0.984i)61-s + (4 − 6.92i)64-s + (3.83 + 3.21i)67-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)4-s + (−0.0656 − 0.372i)7-s + (1.06 + 0.891i)13-s + (−0.939 − 0.342i)16-s + (0.802 + 1.39i)19-s + (−0.766 + 0.642i)25-s + 0.377·28-s + (−0.124 + 0.707i)31-s + (−0.904 + 1.56i)37-s + (−1.14 − 0.417i)43-s + (0.805 − 0.293i)49-s + (−1.06 + 0.891i)52-s + (−0.0222 − 0.126i)61-s + (0.5 − 0.866i)64-s + (0.467 + 0.392i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.899832 + 0.953766i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.899832 + 0.953766i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.173 + 0.984i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.83 - 3.21i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.694 - 3.93i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (7.51 + 2.73i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.173 + 0.984i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.83 - 3.21i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.0 + 10.9i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-17.8 - 6.49i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.61513327210874298667115020417, −9.697064431151749629723910901010, −8.773172659361108804966851889868, −8.088149586737844897680918955967, −7.20620617462523893619729132217, −6.35789006419460820558980929749, −5.11962502753319739703464465246, −3.89218635874035959723193267001, −3.36275353103461006961630319987, −1.65923592381617382130211192189,
0.71330027533948425462725648783, 2.23758673685542541862360751672, 3.61496372560854211377731395538, 4.90893828520675537275892166809, 5.69174325100426903541388450020, 6.43692348135973875668567307408, 7.56559368398209851824218252662, 8.658286509471773915480666684529, 9.321743862294825133484861837392, 10.18587568191887176973264987334