L(s) = 1 | + (−0.988 + 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.900 + 0.433i)8-s + (0.988 − 0.149i)11-s + (0.365 − 0.930i)13-s + (0.826 − 0.563i)16-s + (0.222 + 0.974i)17-s − 19-s + (−0.955 + 0.294i)22-s + (0.955 − 0.294i)23-s + (−0.222 + 0.974i)26-s + (−0.955 − 0.294i)29-s + (0.5 + 0.866i)31-s + (−0.733 + 0.680i)32-s + (−0.365 − 0.930i)34-s + ⋯ |
L(s) = 1 | + (−0.988 + 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.900 + 0.433i)8-s + (0.988 − 0.149i)11-s + (0.365 − 0.930i)13-s + (0.826 − 0.563i)16-s + (0.222 + 0.974i)17-s − 19-s + (−0.955 + 0.294i)22-s + (0.955 − 0.294i)23-s + (−0.222 + 0.974i)26-s + (−0.955 − 0.294i)29-s + (0.5 + 0.866i)31-s + (−0.733 + 0.680i)32-s + (−0.365 − 0.930i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.033540333 + 0.3268861348i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.033540333 + 0.3268861348i\) |
\(L(1)\) |
\(\approx\) |
\(0.7734875072 + 0.07398353966i\) |
\(L(1)\) |
\(\approx\) |
\(0.7734875072 + 0.07398353966i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.988 + 0.149i)T \) |
| 11 | \( 1 + (0.988 - 0.149i)T \) |
| 13 | \( 1 + (0.365 - 0.930i)T \) |
| 17 | \( 1 + (0.222 + 0.974i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.955 - 0.294i)T \) |
| 29 | \( 1 + (-0.955 - 0.294i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.222 + 0.974i)T \) |
| 41 | \( 1 + (0.826 + 0.563i)T \) |
| 43 | \( 1 + (-0.826 + 0.563i)T \) |
| 47 | \( 1 + (0.988 - 0.149i)T \) |
| 53 | \( 1 + (-0.222 + 0.974i)T \) |
| 59 | \( 1 + (0.0747 + 0.997i)T \) |
| 61 | \( 1 + (-0.955 - 0.294i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.623 - 0.781i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.365 - 0.930i)T \) |
| 89 | \( 1 + (0.623 - 0.781i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−19.4899355641366618417705237944, −18.8985424328080043067873738613, −18.403105060804153110710143978982, −17.34993110113164297578029557665, −16.957143565251040997947244590601, −16.28891088751108119417192283133, −15.469997415443047947184968867640, −14.719138797070259823374153074799, −13.96669037911442010860658092180, −12.92786305931848002196272968467, −12.16926888569428685863136364085, −11.3436291391366640683532470928, −11.01971876947930994112279305132, −9.89155499443274059252912595680, −9.204371892575450888940627286963, −8.83889862115546261235535310263, −7.77387426675825809846867006516, −6.99435170556573506930263767687, −6.473813809560702865626680592191, −5.49698485952365029659709262942, −4.252901488270973128431336848297, −3.534861204032648617094051614309, −2.40195975011563187715867983218, −1.68378367650233959163819822020, −0.64228508855543584830287369401,
0.91610448423908410453498538108, 1.63393416996378078655244488172, 2.7671553070422328935415111221, 3.57622105750796584626842422856, 4.6728305743226171517385813766, 5.90169726207612604628276997946, 6.29922135817226357912544997392, 7.202803709638514864647798692936, 8.08665939457456427720394403340, 8.66737315664595050943847933035, 9.34240009140848916218316494903, 10.30519583834650348012415871107, 10.79056259109916728344040825726, 11.562341297511216372482788473776, 12.430172952481969764913167209460, 13.10980423124146391596054550008, 14.245324673226953734510696338281, 15.078543841461941711493116546526, 15.33042003882964269151019952631, 16.51973563102782791838784264130, 16.987855139665652937451245563705, 17.51751596107698367338877822662, 18.412207367253048698455610512354, 19.05038294799214290980005886314, 19.68320040814758265952215619931