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
−19.930222495254509164800277873204, −19.25855898612132915408532662403, −18.859867812414009582672247350272, −17.63255307187789720989331542968, −16.993308497275367522855706340609, −16.14826559398359313468375965716, −15.897809578348545918903470694422, −14.67399672452718000727117133259, −14.077549758204925569378386010624, −13.3563865879875855393379505827, −12.48265147558940556706192404204, −11.491435696640056747818181132041, −11.24931630942608522746387288494, −10.23230486589215356275911198891, −9.31911694378500766659079289573, −8.43567938306408419707797766904, −7.885359016030006704324312527089, −6.7818299907806113478596124697, −6.4429549586451715380359155969, −5.01783067922776828132943815809, −4.15248004321545579980615317439, −3.77026778181902444929400472760, −2.64530919455029316205669979078, −1.2453071019085411028878874782, −0.53836040172595738732445359043,
0.6264554808377148522951743270, 1.86173862602397428098707096567, 2.96707270165308394808119886595, 3.52198114937987259733477246807, 4.76322501428931789273996181043, 5.26450196062093855679189855867, 6.490746027984934262533048586186, 7.148390254904454614424633116423, 7.986850497987075858652881577623, 8.769158584340839661744083541581, 9.51115785832504356816489503662, 10.41681914999669940564358308362, 11.37573428793350179485753975043, 11.96461108584196735221900651939, 12.49495072791142529254316901118, 13.44561901770215099944480854013, 14.43816375934241774260973326417, 15.18650574020497765201306831638, 15.70059510461458380351249574876, 16.16705306817319138400890704398, 17.6964365250507380662011996806, 17.767572472180900331018886770161, 18.78215559782545830616879874791, 19.58544733564663888184202579116, 20.03375374905338669479691600045