L(s) = 1 | + 12i·3-s − 54i·5-s − 88·7-s + 99·9-s + 540i·11-s − 418i·13-s + 648·15-s + 594·17-s − 836i·19-s − 1.05e3i·21-s − 4.10e3·23-s + 209·25-s + 4.10e3i·27-s − 594i·29-s − 4.25e3·31-s + ⋯ |
L(s) = 1 | + 0.769i·3-s − 0.965i·5-s − 0.678·7-s + 0.407·9-s + 1.34i·11-s − 0.685i·13-s + 0.743·15-s + 0.498·17-s − 0.531i·19-s − 0.522i·21-s − 1.61·23-s + 0.0668·25-s + 1.08i·27-s − 0.131i·29-s − 0.795·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4324853237\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4324853237\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 12iT - 243T^{2} \) |
| 5 | \( 1 + 54iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 88T + 1.68e4T^{2} \) |
| 11 | \( 1 - 540iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 418iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 594T + 1.41e6T^{2} \) |
| 19 | \( 1 + 836iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 4.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 594iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 298iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.72e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.21e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.94e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.66e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.47e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.18e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.75e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.77e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 2.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.22e5T + 8.58e9T^{2} \) |
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
−10.42232115203152666088650875087, −9.869342503231410947193766446648, −9.126851443848936814166943575674, −7.939785486910255471890727563564, −6.81986702064647802816185123443, −5.38101408225905076987449756363, −4.56262939147201569380288848886, −3.51872166651258097621825353372, −1.76578556463569406586695507691, −0.11909709289723749298048491877,
1.47415393542951126521269706345, 2.88284002834170367663423348140, 3.92466635453073841614723316181, 5.85997406064744443370246341093, 6.54594806974167425351621693991, 7.42877401219022593199367331278, 8.444285807094076060195889002243, 9.752167845702409155893436600803, 10.55201879853308613260439031575, 11.60541226325730901937735502807