| L(s) = 1 | + (−34.9 + 73.0i)3-s + (−282. − 162. i)5-s + (−2.40e3 − 34.7i)7-s + (−4.11e3 − 5.11e3i)9-s + (1.36e4 − 7.89e3i)11-s − 1.93e4·13-s + (2.17e4 − 1.49e4i)15-s + (8.05e4 − 4.65e4i)17-s + (−5.27e4 + 9.14e4i)19-s + (8.65e4 − 1.74e5i)21-s + (−6.54e4 − 3.78e4i)23-s + (−1.42e5 − 2.46e5i)25-s + (5.17e5 − 1.21e5i)27-s + 4.54e5i·29-s + (3.74e5 + 6.48e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.432 + 0.901i)3-s + (−0.451 − 0.260i)5-s + (−0.999 − 0.0144i)7-s + (−0.626 − 0.779i)9-s + (0.934 − 0.539i)11-s − 0.675·13-s + (0.429 − 0.294i)15-s + (0.964 − 0.557i)17-s + (−0.405 + 0.701i)19-s + (0.445 − 0.895i)21-s + (−0.234 − 0.135i)23-s + (−0.364 − 0.630i)25-s + (0.973 − 0.228i)27-s + 0.642i·29-s + (0.405 + 0.702i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.05190 + 0.354034i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05190 + 0.354034i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (34.9 - 73.0i)T \) |
| 7 | \( 1 + (2.40e3 + 34.7i)T \) |
| good | 5 | \( 1 + (282. + 162. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-1.36e4 + 7.89e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 1.93e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-8.05e4 + 4.65e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (5.27e4 - 9.14e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (6.54e4 + 3.78e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 4.54e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-3.74e5 - 6.48e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.21e6 - 2.11e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.22e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.00e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-4.46e6 - 2.57e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-9.67e6 + 5.58e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.69e7 + 9.78e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (6.75e6 - 1.16e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-8.48e6 - 1.47e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 7.35e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (2.35e6 + 4.08e6i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-7.14e6 + 1.23e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 1.20e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-3.45e7 - 1.99e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.04e8T + 7.83e15T^{2} \) |
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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
−12.34360581421406922173322732658, −11.82424692581649490310711536013, −10.38518351263846342837431316210, −9.592474369604717119352373273733, −8.472364637743558969582991598510, −6.77591804930196208494218672509, −5.61154576809284565101540703013, −4.19030164660863422144675094265, −3.15794251830484652246314267266, −0.67506125061053986003049498111,
0.61506369292647017072677303863, 2.28479626965557089018041972218, 3.87751742681341539087217822297, 5.69972749638830020315757203116, 6.82412783867406896980424677216, 7.62736941812001447751713928422, 9.176579119566739968270822105153, 10.41334428431693158831054335675, 11.76523496193667374235872844553, 12.38023650388436924648127332360
