| L(s) = 1 | + (0.933 + 1.06i)2-s + (−1.40 − 1.01i)3-s + (−0.258 + 1.98i)4-s + (2.52 + 0.501i)5-s + (−0.230 − 2.43i)6-s + (0.531 + 0.220i)7-s + (−2.34 + 1.57i)8-s + (0.938 + 2.84i)9-s + (1.82 + 3.14i)10-s + (2.56 − 1.71i)11-s + (2.37 − 2.52i)12-s + (1.02 + 5.13i)13-s + (0.262 + 0.770i)14-s + (−3.03 − 3.26i)15-s + (−3.86 − 1.02i)16-s + (−2.91 − 2.91i)17-s + ⋯ |
| L(s) = 1 | + (0.659 + 0.751i)2-s + (−0.810 − 0.586i)3-s + (−0.129 + 0.991i)4-s + (1.12 + 0.224i)5-s + (−0.0942 − 0.995i)6-s + (0.200 + 0.0832i)7-s + (−0.830 + 0.557i)8-s + (0.312 + 0.949i)9-s + (0.575 + 0.995i)10-s + (0.772 − 0.515i)11-s + (0.685 − 0.727i)12-s + (0.283 + 1.42i)13-s + (0.0700 + 0.205i)14-s + (−0.782 − 0.842i)15-s + (−0.966 − 0.255i)16-s + (−0.706 − 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29523 + 0.747698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29523 + 0.747698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.933 - 1.06i)T \) |
| 3 | \( 1 + (1.40 + 1.01i)T \) |
| good | 5 | \( 1 + (-2.52 - 0.501i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (-0.531 - 0.220i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-2.56 + 1.71i)T + (4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-1.02 - 5.13i)T + (-12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (2.91 + 2.91i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.478 + 2.40i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.232 + 0.0963i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-5.60 + 8.39i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + 8.55T + 31T^{2} \) |
| 37 | \( 1 + (-0.160 - 0.0319i)T + (34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (1.65 + 3.99i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (3.71 - 2.48i)T + (16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (-0.621 + 0.621i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.49 - 3.73i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (1.43 - 7.22i)T + (-54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (-1.99 + 2.98i)T + (-23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (-10.2 - 6.86i)T + (25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (-5.15 + 12.4i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-3.83 - 9.27i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (6.48 - 6.48i)T - 79iT^{2} \) |
| 83 | \( 1 + (11.1 - 2.21i)T + (76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (6.98 - 16.8i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 14.1iT - 97T^{2} \) |
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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
−12.93563281719457972208254824796, −11.72979023236416187160678679209, −11.22273635802900061220338989828, −9.582385823510411461815582264658, −8.556464993599059253985671735777, −6.99240013445992660018660529820, −6.46233399410283108985743407246, −5.54078150422022647895781409987, −4.33105134123225570073402762212, −2.18620515530704191611791556077,
1.55771893266292780924538722750, 3.55321507628221067552940462978, 4.88546817057100863412829223300, 5.72247363962089493793761678788, 6.61853455750331606004085932542, 8.828020812260320007832084741133, 9.846410895821147660217462002456, 10.47930290919316908943183766931, 11.29855774181820855725938354705, 12.56900860564063865289244692337
