| L(s) = 1 | + (4.74 − 3.44i)2-s + (−28.9 + 88.9i)4-s + (314. + 228. i)5-s + (−446. + 1.37e3i)7-s + (401. + 1.23e3i)8-s + 2.28e3·10-s + (3.72e3 − 2.37e3i)11-s + (−9.08e3 + 6.59e3i)13-s + (2.62e3 + 8.06e3i)14-s + (−3.51e3 − 2.55e3i)16-s + (−1.97e4 − 1.43e4i)17-s + (−8.56e3 − 2.63e4i)19-s + (−2.94e4 + 2.13e4i)20-s + (9.50e3 − 2.41e4i)22-s + 5.52e4·23-s + ⋯ |
| L(s) = 1 | + (0.419 − 0.304i)2-s + (−0.225 + 0.695i)4-s + (1.12 + 0.818i)5-s + (−0.492 + 1.51i)7-s + (0.277 + 0.854i)8-s + 0.722·10-s + (0.843 − 0.537i)11-s + (−1.14 + 0.833i)13-s + (0.255 + 0.785i)14-s + (−0.214 − 0.155i)16-s + (−0.972 − 0.706i)17-s + (−0.286 − 0.881i)19-s + (−0.822 + 0.597i)20-s + (0.190 − 0.482i)22-s + 0.946·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.731126 + 1.96199i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.731126 + 1.96199i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-3.72e3 + 2.37e3i)T \) |
| good | 2 | \( 1 + (-4.74 + 3.44i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-314. - 228. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (446. - 1.37e3i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (9.08e3 - 6.59e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (1.97e4 + 1.43e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (8.56e3 + 2.63e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 5.52e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-2.04e4 + 6.29e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (4.59e4 - 3.33e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (3.61e4 - 1.11e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (-961. - 2.95e3i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 9.90e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.72e5 - 8.39e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (1.36e6 - 9.88e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (1.07e5 - 3.31e5i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-1.31e6 - 9.58e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + 3.78e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (1.99e6 + 1.44e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-8.80e4 + 2.71e5i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (5.74e6 - 4.17e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-2.93e6 - 2.13e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 - 7.93e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-5.53e6 + 4.02e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.89380441024522452628138752597, −11.92581570612687204075096771644, −11.07611380330131208081645819666, −9.302746692164943027945635652530, −9.035993482895144037548593313191, −7.04832466978241186671520574815, −6.03925744502218746135009924148, −4.68208814816678820350365759632, −2.82016650380763633304991885367, −2.33076407090946808389579027908,
0.54181242222053128486732190824, 1.68804510901520871126895470950, 4.02862189325612524162948654487, 5.04369588516022466221461106463, 6.24093313139664767417981352175, 7.24075911864769075247740914280, 9.107024173591842830240298614648, 9.951652544443140782605209693061, 10.61881582879494871588329874161, 12.71430117843656437818157578418
