| L(s) = 1 | + (−1.46 + 1.23i)5-s + (−3.86 + 1.40i)7-s + (−4.34 − 3.64i)11-s + (−0.251 − 1.42i)13-s + (−1.36 − 2.37i)17-s + (−2.54 + 4.40i)19-s + (4.42 + 1.61i)23-s + (−0.229 + 1.29i)25-s + (−0.768 + 4.35i)29-s + (−1.06 − 0.387i)31-s + (3.94 − 6.82i)35-s + (1.97 + 3.41i)37-s + (0.0493 + 0.280i)41-s + (−4.90 − 4.11i)43-s + (−5.79 + 2.10i)47-s + ⋯ |
| L(s) = 1 | + (−0.657 + 0.551i)5-s + (−1.45 + 0.531i)7-s + (−1.31 − 1.10i)11-s + (−0.0696 − 0.394i)13-s + (−0.332 − 0.575i)17-s + (−0.584 + 1.01i)19-s + (0.922 + 0.335i)23-s + (−0.0458 + 0.259i)25-s + (−0.142 + 0.809i)29-s + (−0.191 − 0.0696i)31-s + (0.666 − 1.15i)35-s + (0.324 + 0.562i)37-s + (0.00771 + 0.0437i)41-s + (−0.747 − 0.627i)43-s + (−0.845 + 0.307i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00763664 + 0.160710i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00763664 + 0.160710i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.46 - 1.23i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (3.86 - 1.40i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (4.34 + 3.64i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.251 + 1.42i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.36 + 2.37i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.54 - 4.40i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.42 - 1.61i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.768 - 4.35i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.06 + 0.387i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.97 - 3.41i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0493 - 0.280i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (4.90 + 4.11i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (5.79 - 2.10i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 8.16T + 53T^{2} \) |
| 59 | \( 1 + (10.8 - 9.14i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.76 + 1.00i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.14 + 12.1i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.15 - 5.46i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.36 + 4.09i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.09 - 6.18i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.82 + 16.0i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (1.02 - 1.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.00867 - 0.00727i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.08621111165127864012355183946, −11.02289968923645105072217770808, −10.34960145109949893549945103376, −9.270116496721624907979237902571, −8.271106570026205443950778388104, −7.27225863752316640341729902318, −6.23548744739653822516848424783, −5.28084250428352678073389930232, −3.46769214691458970048820819785, −2.86051530033250353722836444625,
0.10630658011156196489116960784, 2.57820000255361723023583255786, 4.01701679041647880573451002136, 4.92882655134109656242330954458, 6.46680500665733871836716044825, 7.25362872576203213280222557503, 8.301482615478199467619919604742, 9.393257101596175008060394591962, 10.18255007297048873402274129696, 11.08429192381066853227370266806
