| L(s) = 1 | + 1.41i·2-s − 2.00·4-s − 5.19i·5-s + 6.34·7-s − 2.82i·8-s + 7.34·10-s − 9.43i·11-s + 19.6·13-s + 8.97i·14-s + 4.00·16-s − 1.90i·17-s + 4.69·19-s + 10.3i·20-s + 13.3·22-s + 9.43i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.03i·5-s + 0.906·7-s − 0.353i·8-s + 0.734·10-s − 0.858i·11-s + 1.51·13-s + 0.641i·14-s + 0.250·16-s − 0.112i·17-s + 0.247·19-s + 0.519i·20-s + 0.606·22-s + 0.410i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.53997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53997\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.19iT - 25T^{2} \) |
| 7 | \( 1 - 6.34T + 49T^{2} \) |
| 11 | \( 1 + 9.43iT - 121T^{2} \) |
| 13 | \( 1 - 19.6T + 169T^{2} \) |
| 17 | \( 1 + 1.90iT - 289T^{2} \) |
| 19 | \( 1 - 4.69T + 361T^{2} \) |
| 23 | \( 1 - 9.43iT - 529T^{2} \) |
| 29 | \( 1 - 3.28iT - 841T^{2} \) |
| 31 | \( 1 + 41.0T + 961T^{2} \) |
| 37 | \( 1 - 17.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 61.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 0.954T + 1.84e3T^{2} \) |
| 47 | \( 1 - 14.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.53iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 91.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 75.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 30.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 29.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 87.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 95.8T + 9.40e3T^{2} \) |
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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
−12.88625701652401042044577797250, −11.62935106744404132143347949859, −10.71807965055971911841354749621, −9.050836290853121295207364771755, −8.579326368995866687649677907975, −7.56024006225782581850414484474, −5.99430522267691183241930355741, −5.12828783582401275524614668286, −3.82741060615451024193347292959, −1.17591545205960433921518015879,
1.74040861290056030684881170202, 3.30351861002017920098741791142, 4.63858833647485661045741079928, 6.17530538694545667156514045317, 7.48410072990947499629169412342, 8.591780607957441458827971822548, 9.836015677942789114138810782421, 10.93479910200488519871444147303, 11.27611680784888782226348080753, 12.53944457443614707081511184062
