| L(s) = 1 | + (−15.4 − 2.16i)3-s − 81.6i·5-s − 91.7i·7-s + (233. + 66.7i)9-s − 710.·11-s + 666.·13-s + (−176. + 1.25e3i)15-s + 2.17e3i·17-s + 1.12e3i·19-s + (−198. + 1.41e3i)21-s + 3.12e3·23-s − 3.53e3·25-s + (−3.46e3 − 1.53e3i)27-s + 371. i·29-s − 2.50e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.990 − 0.138i)3-s − 1.45i·5-s − 0.707i·7-s + (0.961 + 0.274i)9-s − 1.77·11-s + 1.09·13-s + (−0.202 + 1.44i)15-s + 1.82i·17-s + 0.715i·19-s + (−0.0981 + 0.700i)21-s + 1.22·23-s − 1.13·25-s + (−0.914 − 0.405i)27-s + 0.0819i·29-s − 0.468i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.079980336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079980336\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (15.4 + 2.16i)T \) |
| good | 5 | \( 1 + 81.6iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 91.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 710.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 666.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.17e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.12e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 371. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.50e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 6.27e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.54e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.15e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.36e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.37e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.47e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.27e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 2.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.87e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e4T + 8.58e9T^{2} \) |
| 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
−10.61532989502731481894584697534, −9.811667471790694659089555796162, −8.364911731830535458069829423310, −7.939334811801586645889720102024, −6.51323563400858591715864654201, −5.52096956717940814547364589467, −4.82667710865994123137999088650, −3.76020976922781629906611269144, −1.62226913747908931082707703939, −0.76326734775177062580375152605,
0.45727613433793779834756244518, 2.41586841863542944537803133007, 3.27782938378026345645601087393, 4.99971936661710017881725549942, 5.63346438598156994708430196743, 6.84064496908667311742467119645, 7.32316155603268230131559787187, 8.777154033207282646676025440459, 9.937165431151966524753517322992, 10.82531591403829048163160150446
