L(s) = 1 | − 2-s + (0.686 + 0.727i)3-s + 4-s + (−0.549 − 0.835i)5-s + (−0.686 − 0.727i)6-s + (−0.0581 − 0.998i)7-s − 8-s + (−0.0581 + 0.998i)9-s + (0.549 + 0.835i)10-s + (−0.549 + 0.835i)11-s + (0.686 + 0.727i)12-s + (0.835 − 0.549i)13-s + (0.0581 + 0.998i)14-s + (0.230 − 0.973i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | − 2-s + (0.686 + 0.727i)3-s + 4-s + (−0.549 − 0.835i)5-s + (−0.686 − 0.727i)6-s + (−0.0581 − 0.998i)7-s − 8-s + (−0.0581 + 0.998i)9-s + (0.549 + 0.835i)10-s + (−0.549 + 0.835i)11-s + (0.686 + 0.727i)12-s + (0.835 − 0.549i)13-s + (0.0581 + 0.998i)14-s + (0.230 − 0.973i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1014193319 - 0.2783902023i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1014193319 - 0.2783902023i\) |
\(L(1)\) |
\(\approx\) |
\(0.6581040212 + 0.008437261544i\) |
\(L(1)\) |
\(\approx\) |
\(0.6581040212 + 0.008437261544i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.686 + 0.727i)T \) |
| 5 | \( 1 + (-0.549 - 0.835i)T \) |
| 7 | \( 1 + (-0.0581 - 0.998i)T \) |
| 11 | \( 1 + (-0.549 + 0.835i)T \) |
| 13 | \( 1 + (0.835 - 0.549i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.766 - 0.642i)T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.998 - 0.0581i)T \) |
| 31 | \( 1 + (0.918 + 0.396i)T \) |
| 41 | \( 1 + (0.342 - 0.939i)T \) |
| 43 | \( 1 + (-0.642 - 0.766i)T \) |
| 47 | \( 1 + (0.727 + 0.686i)T \) |
| 53 | \( 1 + (0.957 + 0.286i)T \) |
| 59 | \( 1 + (-0.973 - 0.230i)T \) |
| 61 | \( 1 + (0.116 - 0.993i)T \) |
| 67 | \( 1 + (0.727 + 0.686i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.0581 - 0.998i)T \) |
| 79 | \( 1 + (-0.973 - 0.230i)T \) |
| 83 | \( 1 + (-0.597 + 0.802i)T \) |
| 89 | \( 1 + (0.448 - 0.893i)T \) |
| 97 | \( 1 + (-0.918 + 0.396i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.79613257480099790231575118027, −18.125301838593646942487218181439, −17.964706922044315288385261992, −16.61744961147563703837428663111, −15.931065129104070461356250624599, −15.47593680157837664025291852146, −14.768429911585578796700948432814, −14.0676208246253633662182793960, −13.32862479346428136256149147732, −12.315313827676429468812558705769, −11.74011493004332850462595358532, −11.28104924344867588229901735395, −10.36533187698672296244610296555, −9.65558368055595200803427403102, −8.61256703388266812244206938417, −8.455837274599405959902833103361, −7.80227452889145855469655806367, −6.880902517509513304445402177303, −6.297979099526692077379907833050, −5.859850968168734336871671812557, −4.24854325329072045466128937738, −3.25935637102313566682709448291, −2.65106716028933175440987687828, −2.18147514229162515794030078723, −1.08222147042823100348208708764,
0.112724666919900944121248814421, 1.26124035687788925390261260751, 2.08672120471872153289965786928, 3.044481141944500168111959943916, 3.97127742679769342781780730252, 4.40288915954499294016961186793, 5.39607511958577850534476356084, 6.44382920348260692100695062393, 7.347381494076356270472899688630, 7.99296021336836327316192892297, 8.46086515057705102138568878504, 9.05070822155651848170532327434, 9.96966628601594876921356919489, 10.48223799308609384910335713805, 10.90403355336891642020464009043, 11.91338190951349166127617715086, 12.76127770398440621443495547902, 13.36732384905557865175563810287, 14.214927707948074268033920818795, 15.28258816349025078683726477792, 15.63019848686255686578835783885, 15.98940141696511801165424714321, 17.01225271112288582098639252994, 17.31329952524697205211955731306, 18.08750558455971387012043537018