| L(s) = 1 | + (5.16 + 14.7i)3-s + 48.6i·5-s + 167. i·7-s + (−189. + 151. i)9-s + 319.·11-s − 152.·13-s + (−715. + 251. i)15-s + 1.32e3i·17-s + 1.61e3i·19-s + (−2.46e3 + 865. i)21-s + 1.22e3·23-s + 756.·25-s + (−3.21e3 − 2.00e3i)27-s + 4.54e3i·29-s + 5.35e3i·31-s + ⋯ |
| L(s) = 1 | + (0.331 + 0.943i)3-s + 0.870i·5-s + 1.29i·7-s + (−0.780 + 0.624i)9-s + 0.795·11-s − 0.250·13-s + (−0.821 + 0.288i)15-s + 1.11i·17-s + 1.02i·19-s + (−1.22 + 0.428i)21-s + 0.481·23-s + 0.242·25-s + (−0.848 − 0.529i)27-s + 1.00i·29-s + 1.00i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.111724097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.111724097\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-5.16 - 14.7i)T \) |
| good | 5 | \( 1 - 48.6iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 167. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 319.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 152.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.32e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.61e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.54e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.35e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 3.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.30e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 2.04e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.32e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 5.88e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.39e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.22e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 1.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.70e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.59e5T + 8.58e9T^{2} \) |
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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
−10.70711269354708162120124738975, −10.31068734054815758851243528358, −8.950422071689575131983020081088, −8.730686608789439677610403938593, −7.28718942422949055091484455789, −6.09969590954283388450566303259, −5.28646904055101715112778002547, −3.90242274397393455765867106784, −3.03445243403703645956247354873, −1.90673888713853306636243465750,
0.56295526999231804046149506867, 1.10671507956286573078553947358, 2.62475946819587153491599589486, 4.01165547411217109708962966120, 5.02180743049959230317554047065, 6.47952858915965661194209973139, 7.19988373489809122853954202581, 8.036373122061496694195774397050, 9.074956592776559305027763223404, 9.749192226776135016951139257320
