| L(s) = 1 | − 3.61i·5-s − 9.61·7-s + 3.31i·11-s − 6.68·13-s − 21.3i·17-s + 18.2·19-s − 18.4i·23-s + 11.9·25-s + 42.0i·29-s − 27.9·31-s + 34.7i·35-s − 27.1·37-s + 70.5i·41-s − 59.6·43-s + 74.8i·47-s + ⋯ |
| L(s) = 1 | − 0.723i·5-s − 1.37·7-s + 0.301i·11-s − 0.513·13-s − 1.25i·17-s + 0.959·19-s − 0.804i·23-s + 0.476·25-s + 1.44i·29-s − 0.902·31-s + 0.993i·35-s − 0.733·37-s + 1.72i·41-s − 1.38·43-s + 1.59i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1188 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1188 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6913740956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6913740956\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - 3.31iT \) |
| good | 5 | \( 1 + 3.61iT - 25T^{2} \) |
| 7 | \( 1 + 9.61T + 49T^{2} \) |
| 13 | \( 1 + 6.68T + 169T^{2} \) |
| 17 | \( 1 + 21.3iT - 289T^{2} \) |
| 19 | \( 1 - 18.2T + 361T^{2} \) |
| 23 | \( 1 + 18.4iT - 529T^{2} \) |
| 29 | \( 1 - 42.0iT - 841T^{2} \) |
| 31 | \( 1 + 27.9T + 961T^{2} \) |
| 37 | \( 1 + 27.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 70.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 59.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 74.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 83.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 97.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 107.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 70.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 53.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 55.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 125.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 129. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 98.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 61.4T + 9.40e3T^{2} \) |
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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
−9.567937050680841680044065536309, −9.217879445330791185083481864757, −8.218345154469584024801734282546, −7.09189617517473567406009349812, −6.65904331388199096627651345414, −5.35625074103485005108131034518, −4.80560003360169105296180545292, −3.48426122893797435977930668022, −2.67587871925669947151609063182, −1.05078076656330568088439309214,
0.23252889864296892192992352636, 2.04858630308388322320207200382, 3.27821945438789244257630884155, 3.73038812979136303926060650366, 5.27855959664889765463667112511, 6.10453804763637057651089865494, 6.85668690032420964541609731015, 7.54316144162477138331355556732, 8.629614260373032330876500398724, 9.521806818596154454514053955896
