L(s) = 1 | + (−0.759 − 0.650i)2-s + (0.00620 + 0.999i)3-s + (0.154 + 0.987i)4-s + (−0.471 + 0.882i)5-s + (0.645 − 0.763i)6-s + (−0.426 + 0.904i)7-s + (0.524 − 0.851i)8-s + (−0.999 + 0.0124i)9-s + (0.931 − 0.363i)10-s + (−0.311 + 0.950i)11-s + (−0.986 + 0.160i)12-s + (−0.492 + 0.870i)13-s + (0.912 − 0.409i)14-s + (−0.885 − 0.465i)15-s + (−0.952 + 0.305i)16-s + (−0.996 − 0.0868i)17-s + ⋯ |
L(s) = 1 | + (−0.759 − 0.650i)2-s + (0.00620 + 0.999i)3-s + (0.154 + 0.987i)4-s + (−0.471 + 0.882i)5-s + (0.645 − 0.763i)6-s + (−0.426 + 0.904i)7-s + (0.524 − 0.851i)8-s + (−0.999 + 0.0124i)9-s + (0.931 − 0.363i)10-s + (−0.311 + 0.950i)11-s + (−0.986 + 0.160i)12-s + (−0.492 + 0.870i)13-s + (0.912 − 0.409i)14-s + (−0.885 − 0.465i)15-s + (−0.952 + 0.305i)16-s + (−0.996 − 0.0868i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1102173061 + 0.3095919740i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1102173061 + 0.3095919740i\) |
\(L(1)\) |
\(\approx\) |
\(0.4367554419 + 0.2702240215i\) |
\(L(1)\) |
\(\approx\) |
\(0.4367554419 + 0.2702240215i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.759 - 0.650i)T \) |
| 3 | \( 1 + (0.00620 + 0.999i)T \) |
| 5 | \( 1 + (-0.471 + 0.882i)T \) |
| 7 | \( 1 + (-0.426 + 0.904i)T \) |
| 11 | \( 1 + (-0.311 + 0.950i)T \) |
| 13 | \( 1 + (-0.492 + 0.870i)T \) |
| 17 | \( 1 + (-0.996 - 0.0868i)T \) |
| 19 | \( 1 + (-0.287 - 0.957i)T \) |
| 29 | \( 1 + (0.251 + 0.967i)T \) |
| 31 | \( 1 + (0.813 + 0.581i)T \) |
| 37 | \( 1 + (-0.673 + 0.739i)T \) |
| 41 | \( 1 + (0.783 - 0.621i)T \) |
| 43 | \( 1 + (0.626 - 0.779i)T \) |
| 47 | \( 1 + (0.460 + 0.887i)T \) |
| 53 | \( 1 + (0.931 + 0.363i)T \) |
| 59 | \( 1 + (0.751 + 0.659i)T \) |
| 61 | \( 1 + (0.940 + 0.340i)T \) |
| 67 | \( 1 + (-0.993 + 0.111i)T \) |
| 71 | \( 1 + (-0.0434 - 0.999i)T \) |
| 73 | \( 1 + (0.251 - 0.967i)T \) |
| 79 | \( 1 + (0.105 - 0.994i)T \) |
| 83 | \( 1 + (0.0806 + 0.996i)T \) |
| 89 | \( 1 + (-0.977 + 0.209i)T \) |
| 97 | \( 1 + (0.323 + 0.946i)T \) |
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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
−23.16522667198678169613705698625, −22.73829137325842276517302203243, −20.906371931162351282114687136627, −20.034156293892482698513476645358, −19.48853036181213074600597677557, −18.86936544231933821216215202224, −17.68782583677344776245291686081, −17.14586924214234128730654712673, −16.36821491149316815692507211973, −15.58361287756280329887226919114, −14.40872380737227663100496511234, −13.454089475603263732050988393520, −12.87488922653280797429849265332, −11.66302863175852713668350314310, −10.7606694442871549723828234847, −9.716626233888044009239972399155, −8.500133160797582321306545663123, −8.04875656055926256131249020147, −7.232469673333220905060007568733, −6.208064272497266147022699324869, −5.4072058854741042346021299081, −4.05230055618144322372789962469, −2.478324916315255721193749992178, −1.04690637355333555985882247765, −0.25907661926225198786515278986,
2.319275972264867501018668700573, 2.778752278288893375446394616290, 4.003591853508876395873966495604, 4.85432779089962417829437978193, 6.51957288210706008173616184463, 7.286137556009011941431561176481, 8.67697519252932475797221246048, 9.19794034970815201171203798312, 10.166959686888111180714544657638, 10.832669105557022464444502300286, 11.74300788552470032498762298533, 12.36647881748777513031535367047, 13.75476187517244305520855470552, 14.99778366400730916027682596080, 15.56788133683405051941392217942, 16.25668706508120396367864421850, 17.49928556316922461322921634809, 18.03000054136306653879733777854, 19.24460634467503549568582535304, 19.57032837217219960177055084683, 20.63697162052856041983639777565, 21.49909503013941248572175920414, 22.23885424306874283052681359670, 22.57067217520517226334851232168, 23.92086695032090977509585273077