L(s) = 1 | + (−7.69 − 8.29i)2-s − 27i·3-s + (−9.58 + 127. i)4-s − 455. i·5-s + (−223. + 207. i)6-s − 743.·7-s + (1.13e3 − 902. i)8-s − 729·9-s + (−3.77e3 + 3.50e3i)10-s + 5.47e3i·11-s + (3.44e3 + 258. i)12-s + 6.21e3i·13-s + (5.72e3 + 6.16e3i)14-s − 1.22e4·15-s + (−1.62e4 − 2.44e3i)16-s − 2.63e4·17-s + ⋯ |
L(s) = 1 | + (−0.680 − 0.733i)2-s − 0.577i·3-s + (−0.0748 + 0.997i)4-s − 1.62i·5-s + (−0.423 + 0.392i)6-s − 0.819·7-s + (0.781 − 0.623i)8-s − 0.333·9-s + (−1.19 + 1.10i)10-s + 1.24i·11-s + (0.575 + 0.0432i)12-s + 0.785i·13-s + (0.557 + 0.600i)14-s − 0.940·15-s + (−0.988 − 0.149i)16-s − 1.29·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.170018 + 0.352948i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.170018 + 0.352948i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (7.69 + 8.29i)T \) |
| 3 | \( 1 + 27iT \) |
good | 5 | \( 1 + 455. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 743.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.47e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 6.21e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 2.63e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.33e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 5.09e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.85e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.84e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.35e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 5.39e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.04e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 9.23e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.21e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.07e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.76e5iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 2.56e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 2.50e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.93e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.77e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 9.69e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.83e6T + 8.07e13T^{2} \) |
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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
−15.77302284381033518333459082332, −13.23002316268990569867812349737, −12.75842207663140142061576044473, −11.56938379910158165632201501826, −9.559502640399605867745831240135, −8.762670746597119490024378822388, −7.05656551291368284395415863526, −4.48033044127846278033921190528, −1.94126982389817703782124886133, −0.24793895676006740432313063532,
3.16466104258851714977611655734, 5.88789838715415080441995117582, 7.05584744742849632703493617629, 8.836845551535806474317344333924, 10.36396977605274413222978430503, 10.98932242238664304281881788078, 13.62945898662348007252230153265, 14.81391273623779667686740136927, 15.65711153118877096346935149480, 16.79214780478813965173637690390