L(s) = 1 | + (−0.976 − 0.216i)5-s + (−0.0871 − 0.996i)7-s + (0.537 − 0.843i)11-s + (0.0436 + 0.999i)13-s + (−0.866 + 0.5i)17-s + (0.991 + 0.130i)19-s + (0.0871 − 0.996i)23-s + (0.906 + 0.422i)25-s + (0.737 − 0.675i)29-s + (0.766 − 0.642i)31-s + (−0.130 + 0.991i)35-s + (−0.991 + 0.130i)37-s + (0.906 − 0.422i)41-s + (−0.537 + 0.843i)43-s + (−0.642 + 0.766i)47-s + ⋯ |
L(s) = 1 | + (−0.976 − 0.216i)5-s + (−0.0871 − 0.996i)7-s + (0.537 − 0.843i)11-s + (0.0436 + 0.999i)13-s + (−0.866 + 0.5i)17-s + (0.991 + 0.130i)19-s + (0.0871 − 0.996i)23-s + (0.906 + 0.422i)25-s + (0.737 − 0.675i)29-s + (0.766 − 0.642i)31-s + (−0.130 + 0.991i)35-s + (−0.991 + 0.130i)37-s + (0.906 − 0.422i)41-s + (−0.537 + 0.843i)43-s + (−0.642 + 0.766i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.545414989 - 0.6968347582i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.545414989 - 0.6968347582i\) |
\(L(1)\) |
\(\approx\) |
\(0.9363875641 - 0.1672528677i\) |
\(L(1)\) |
\(\approx\) |
\(0.9363875641 - 0.1672528677i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.976 - 0.216i)T \) |
| 7 | \( 1 + (-0.0871 - 0.996i)T \) |
| 11 | \( 1 + (0.537 - 0.843i)T \) |
| 13 | \( 1 + (0.0436 + 0.999i)T \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.991 + 0.130i)T \) |
| 23 | \( 1 + (0.0871 - 0.996i)T \) |
| 29 | \( 1 + (0.737 - 0.675i)T \) |
| 31 | \( 1 + (0.766 - 0.642i)T \) |
| 37 | \( 1 + (-0.991 + 0.130i)T \) |
| 41 | \( 1 + (0.906 - 0.422i)T \) |
| 43 | \( 1 + (-0.537 + 0.843i)T \) |
| 47 | \( 1 + (-0.642 + 0.766i)T \) |
| 53 | \( 1 + (-0.923 - 0.382i)T \) |
| 59 | \( 1 + (0.976 + 0.216i)T \) |
| 61 | \( 1 + (0.953 + 0.300i)T \) |
| 67 | \( 1 + (0.999 - 0.0436i)T \) |
| 71 | \( 1 + (-0.258 + 0.965i)T \) |
| 73 | \( 1 + (0.258 + 0.965i)T \) |
| 79 | \( 1 + (0.342 + 0.939i)T \) |
| 83 | \( 1 + (0.675 + 0.737i)T \) |
| 89 | \( 1 + (0.965 - 0.258i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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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
−20.03375374905338669479691600045, −19.58544733564663888184202579116, −18.78215559782545830616879874791, −17.767572472180900331018886770161, −17.6964365250507380662011996806, −16.16705306817319138400890704398, −15.70059510461458380351249574876, −15.18650574020497765201306831638, −14.43816375934241774260973326417, −13.44561901770215099944480854013, −12.49495072791142529254316901118, −11.96461108584196735221900651939, −11.37573428793350179485753975043, −10.41681914999669940564358308362, −9.51115785832504356816489503662, −8.769158584340839661744083541581, −7.986850497987075858652881577623, −7.148390254904454614424633116423, −6.490746027984934262533048586186, −5.26450196062093855679189855867, −4.76322501428931789273996181043, −3.52198114937987259733477246807, −2.96707270165308394808119886595, −1.86173862602397428098707096567, −0.6264554808377148522951743270,
0.53836040172595738732445359043, 1.2453071019085411028878874782, 2.64530919455029316205669979078, 3.77026778181902444929400472760, 4.15248004321545579980615317439, 5.01783067922776828132943815809, 6.4429549586451715380359155969, 6.7818299907806113478596124697, 7.885359016030006704324312527089, 8.43567938306408419707797766904, 9.31911694378500766659079289573, 10.23230486589215356275911198891, 11.24931630942608522746387288494, 11.491435696640056747818181132041, 12.48265147558940556706192404204, 13.3563865879875855393379505827, 14.077549758204925569378386010624, 14.67399672452718000727117133259, 15.897809578348545918903470694422, 16.14826559398359313468375965716, 16.993308497275367522855706340609, 17.63255307187789720989331542968, 18.859867812414009582672247350272, 19.25855898612132915408532662403, 19.930222495254509164800277873204