| L(s) = 1 | + 2.23·5-s − 8.50i·7-s + 1.79i·11-s − 0.472·13-s + 23.8·17-s + 9.40i·19-s + 16.1i·23-s + 5.00·25-s + 6.94·29-s + 47.4i·31-s − 19.0i·35-s − 26.3·37-s + 41.4·41-s + 2.00i·43-s + 35.3i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.21i·7-s + 0.163i·11-s − 0.0363·13-s + 1.40·17-s + 0.494i·19-s + 0.700i·23-s + 0.200·25-s + 0.239·29-s + 1.53i·31-s − 0.543i·35-s − 0.712·37-s + 1.01·41-s + 0.0467i·43-s + 0.752i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.433891728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.433891728\) |
| \(L(2)\) |
|
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 | \( 1 - 2.23T \) |
| good | 7 | \( 1 + 8.50iT - 49T^{2} \) |
| 11 | \( 1 - 1.79iT - 121T^{2} \) |
| 13 | \( 1 + 0.472T + 169T^{2} \) |
| 17 | \( 1 - 23.8T + 289T^{2} \) |
| 19 | \( 1 - 9.40iT - 361T^{2} \) |
| 23 | \( 1 - 16.1iT - 529T^{2} \) |
| 29 | \( 1 - 6.94T + 841T^{2} \) |
| 31 | \( 1 - 47.4iT - 961T^{2} \) |
| 37 | \( 1 + 26.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 41.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 2.00iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 21.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 73.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 88.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 39.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 21.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 39.1T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.562110671870032976281876994538, −7.66726833411702025045798704661, −7.23912203461672488045596561892, −6.33049228087987003325810595781, −5.51550666195513910513216899984, −4.72133939841906126940701838443, −3.75279665168150709678058266145, −3.06584818607479081829725619591, −1.67404255355737824178401161557, −0.878789508790849021706689474324,
0.72223603003272404893897747041, 2.05710843784728009460605774794, 2.74372562248170285140700762109, 3.74323714214740719587029954357, 4.90707480686834887537401100607, 5.58959072744714975301437868446, 6.14165094424850162581647262051, 7.03932126545157152923035438008, 8.011777332169540436517757799333, 8.568881833015491280098829875761
