| L(s) = 1 | + (−4.82 − 1.32i)5-s + (−5.42 + 3.13i)7-s + (−0.895 + 0.516i)11-s + (2.59 + 1.5i)13-s + 15.7·17-s + 18.7·19-s + (−17.3 + 30.0i)23-s + (21.5 + 12.7i)25-s + (−2.68 + 1.55i)29-s + (8.89 − 15.4i)31-s + (30.2 − 7.93i)35-s + 43.3i·37-s + (−40.0 − 23.1i)41-s + (−17.9 + 10.3i)43-s + (−39.4 − 68.3i)47-s + ⋯ |
| L(s) = 1 | + (−0.964 − 0.264i)5-s + (−0.774 + 0.447i)7-s + (−0.0813 + 0.0469i)11-s + (0.199 + 0.115i)13-s + 0.928·17-s + 0.988·19-s + (−0.753 + 1.30i)23-s + (0.860 + 0.509i)25-s + (−0.0926 + 0.0534i)29-s + (0.286 − 0.496i)31-s + (0.865 − 0.226i)35-s + 1.17i·37-s + (−0.977 − 0.564i)41-s + (−0.417 + 0.241i)43-s + (−0.839 − 1.45i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4488516197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4488516197\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.82 + 1.32i)T \) |
| good | 7 | \( 1 + (5.42 - 3.13i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (0.895 - 0.516i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-2.59 - 1.5i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 15.7T + 289T^{2} \) |
| 19 | \( 1 - 18.7T + 361T^{2} \) |
| 23 | \( 1 + (17.3 - 30.0i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (2.68 - 1.55i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-8.89 + 15.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 43.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (40.0 + 23.1i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (17.9 - 10.3i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (39.4 + 68.3i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 44.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (78.4 + 45.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (4.39 + 7.60i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-16.7 - 9.65i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 56.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 109. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (19.5 + 33.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (37.8 + 65.6i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 90.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (7.69 - 4.44i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−8.942882585179157659286281115096, −8.036235429683763795719521543013, −7.49654853407454170097076579462, −6.53470663917162869088461570365, −5.61732031008892744304628969927, −4.81558073603919906306407634359, −3.57400712426058352372311678453, −3.17372928639200067969345543012, −1.55440723448545700337678952721, −0.14659567361724741184676563585,
1.02569418076263097147125611670, 2.77765185898480095058606221947, 3.51695553231192978093966449513, 4.30753777302920181655785405683, 5.39906950520883525686254845339, 6.41796725524688806217066981852, 7.11199846785664108487436033937, 7.88537744009115800179162852620, 8.526847184250885939546155144531, 9.620662788275739147835925433927
