| L(s) = 1 | + (−13.2 + 9.02i)2-s + (−34.6 − 11.2i)3-s + (92.9 − 238. i)4-s + (617. + 98.5i)5-s + (559. − 164. i)6-s − 4.12e3i·7-s + (926. + 3.98e3i)8-s + (−4.23e3 − 3.07e3i)9-s + (−9.04e3 + 4.27e3i)10-s + (−1.43e4 − 1.98e4i)11-s + (−5.91e3 + 7.22e3i)12-s + (−2.43e4 − 1.76e4i)13-s + (3.72e4 + 5.45e4i)14-s + (−2.02e4 − 1.03e4i)15-s + (−4.82e4 − 4.43e4i)16-s + (−1.33e3 − 4.09e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.825 + 0.564i)2-s + (−0.428 − 0.139i)3-s + (0.363 − 0.931i)4-s + (0.987 + 0.157i)5-s + (0.431 − 0.126i)6-s − 1.71i·7-s + (0.226 + 0.974i)8-s + (−0.645 − 0.468i)9-s + (−0.904 + 0.427i)10-s + (−0.983 − 1.35i)11-s + (−0.285 + 0.348i)12-s + (−0.851 − 0.618i)13-s + (0.970 + 1.41i)14-s + (−0.400 − 0.204i)15-s + (−0.736 − 0.676i)16-s + (−0.0159 − 0.0490i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0510037 + 0.408721i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0510037 + 0.408721i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (13.2 - 9.02i)T \) |
| 5 | \( 1 + (-617. - 98.5i)T \) |
| good | 3 | \( 1 + (34.6 + 11.2i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 + 4.12e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (1.43e4 + 1.98e4i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (2.43e4 + 1.76e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (1.33e3 + 4.09e3i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-3.92e4 + 1.27e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (1.20e5 + 1.65e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (2.49e5 - 7.69e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-1.90e5 + 6.17e4i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (1.30e6 + 9.49e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-3.64e6 - 2.64e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 5.06e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-1.07e6 - 3.49e5i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-2.39e6 + 7.37e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-9.02e6 + 1.24e7i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (9.78e6 - 7.11e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (8.00e6 - 2.60e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-3.59e6 - 1.16e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (3.09e7 - 2.24e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-7.22e7 - 2.34e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (1.15e6 - 3.74e5i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (2.20e7 - 1.60e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (-1.67e7 + 5.16e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−11.10512527595532550401301029432, −10.57123502874918229514620478615, −9.641481443168435550619318177844, −8.263483804410989988995871341683, −7.17146143449174476871866436317, −6.11799225523742302051769881044, −5.15969006496943933931413634645, −2.90148481840806144716357282659, −1.00531721123473911050985078880, −0.18246178728504166785650927898,
2.09809558440768323722644579345, 2.44109456765310742403381252845, 4.93216838247251715691290481155, 5.91952099020993451928690736409, 7.52444297179411249416799263693, 8.841955367277455357826615278113, 9.631266684896214617770644755726, 10.50466909441251477973118127650, 11.92542657575556567379086451908, 12.32275469558915385703023684177
