| L(s) = 1 | + (−0.982 + 2.83i)3-s + (1.05 + 0.611i)5-s + (5.05 − 2.91i)7-s + (−7.07 − 5.56i)9-s + (3.23 + 5.60i)11-s + (7.09 + 4.09i)13-s + (−2.77 + 2.40i)15-s − 2.98·17-s + 22.6·19-s + (3.30 + 17.2i)21-s + (29.2 + 16.8i)23-s + (−11.7 − 20.3i)25-s + (22.7 − 14.5i)27-s + (−11.7 + 6.75i)29-s + (1.96 + 1.13i)31-s + ⋯ |
| L(s) = 1 | + (−0.327 + 0.944i)3-s + (0.211 + 0.122i)5-s + (0.722 − 0.417i)7-s + (−0.785 − 0.618i)9-s + (0.294 + 0.509i)11-s + (0.545 + 0.314i)13-s + (−0.185 + 0.160i)15-s − 0.175·17-s + 1.19·19-s + (0.157 + 0.819i)21-s + (1.27 + 0.734i)23-s + (−0.470 − 0.814i)25-s + (0.841 − 0.539i)27-s + (−0.403 + 0.232i)29-s + (0.0633 + 0.0365i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.804098640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804098640\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.982 - 2.83i)T \) |
| good | 5 | \( 1 + (-1.05 - 0.611i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-5.05 + 2.91i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.23 - 5.60i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-7.09 - 4.09i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 2.98T + 289T^{2} \) |
| 19 | \( 1 - 22.6T + 361T^{2} \) |
| 23 | \( 1 + (-29.2 - 16.8i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (11.7 - 6.75i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-1.96 - 1.13i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 26.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-3.95 + 6.85i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-21.9 - 38.0i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (47.7 - 27.5i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 97.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-24.0 + 41.6i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-17.9 + 10.3i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-11.9 + 20.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 93.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 88.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-38.7 + 22.3i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-30.3 - 52.6i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 120.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (77.3 + 134. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.81656713372849898186836578818, −9.763348533189775633382612694614, −9.230897036458801615545503254462, −8.126090883864986124073838664098, −7.08080001662493292158721491811, −6.00203078189437288148021104863, −4.99328066809338797251898403297, −4.21867009438731721193467486883, −3.06409256532945555688541253381, −1.28790677803361616947029018832,
0.848312880929758667931337248213, 2.02304882141904099064047935673, 3.37757004358973380511985109246, 5.07416360079691613027294067799, 5.68357896247169379692502429808, 6.72656051070810043371886733077, 7.64654007842412364400525012117, 8.498589196347675302420009546352, 9.224326851886291916495491147190, 10.58259660442228795331278066044
