| L(s) = 1 | + (−40.5 − 23.3i)3-s + (−489. + 282. i)5-s + (−1.45e3 + 1.90e3i)7-s + (1.09e3 + 1.89e3i)9-s + (−4.10e3 + 7.11e3i)11-s + 1.36e4i·13-s + 2.64e4·15-s + (6.39e3 + 3.69e3i)17-s + (2.66e4 − 1.53e4i)19-s + (1.03e5 − 4.32e4i)21-s + (−5.74e4 − 9.95e4i)23-s + (−3.53e4 + 6.12e4i)25-s − 1.02e5i·27-s − 1.24e5·29-s + (−3.31e5 − 1.91e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 − 0.288i)3-s + (−0.783 + 0.452i)5-s + (−0.606 + 0.794i)7-s + (0.166 + 0.288i)9-s + (−0.280 + 0.485i)11-s + 0.477i·13-s + 0.522·15-s + (0.0765 + 0.0441i)17-s + (0.204 − 0.118i)19-s + (0.532 − 0.222i)21-s + (−0.205 − 0.355i)23-s + (−0.0905 + 0.156i)25-s − 0.192i·27-s − 0.175·29-s + (−0.358 − 0.207i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00155 + 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.00155 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.292023 - 0.292477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.292023 - 0.292477i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (40.5 + 23.3i)T \) |
| 7 | \( 1 + (1.45e3 - 1.90e3i)T \) |
| good | 5 | \( 1 + (489. - 282. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (4.10e3 - 7.11e3i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 1.36e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-6.39e3 - 3.69e3i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-2.66e4 + 1.53e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (5.74e4 + 9.95e4i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 1.24e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (3.31e5 + 1.91e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (6.78e5 + 1.17e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 2.68e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.62e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.06e6 + 1.77e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-4.54e6 + 7.86e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.97e5 - 1.13e5i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.18e7 + 6.83e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-4.13e6 + 7.16e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.81e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.49e7 - 8.65e6i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.26e7 + 2.18e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 1.68e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-6.75e7 + 3.89e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.50e8iT - 7.83e15T^{2} \) |
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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
−12.21189628549377697119010807127, −11.48643576184439941568793651382, −10.28335255893414452805227353193, −8.996513216049197268034019722687, −7.59504277591061603151168962927, −6.61002510686241585247161468460, −5.31362094939250854858053154180, −3.71724518222661915194007898359, −2.18612507613020022483326025480, −0.17553301396644232853698548192,
0.893717875249388026702384540813, 3.30412504361416699259467679756, 4.42058835656056338614157372890, 5.79831600984332587154526547211, 7.20587623673016403999051186134, 8.331360304559726467282693255735, 9.763232318474116233931971144930, 10.74282915052537080975716832518, 11.79615343050829436043845619786, 12.80351094084796756958178927602
