L(s) = 1 | + (−0.406 + 0.913i)2-s + (0.743 − 0.669i)3-s + (−0.669 − 0.743i)4-s + (0.309 + 0.951i)6-s + (0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.994 − 0.104i)12-s + (−0.587 + 0.809i)13-s + (−0.104 + 0.994i)16-s + (0.207 − 0.978i)17-s + (0.866 + 0.5i)18-s + (−0.669 + 0.743i)19-s + (0.951 + 0.309i)22-s + (0.406 − 0.913i)23-s + (0.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (−0.406 + 0.913i)2-s + (0.743 − 0.669i)3-s + (−0.669 − 0.743i)4-s + (0.309 + 0.951i)6-s + (0.951 − 0.309i)8-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (−0.994 − 0.104i)12-s + (−0.587 + 0.809i)13-s + (−0.104 + 0.994i)16-s + (0.207 − 0.978i)17-s + (0.866 + 0.5i)18-s + (−0.669 + 0.743i)19-s + (0.951 + 0.309i)22-s + (0.406 − 0.913i)23-s + (0.5 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6538322312 - 0.7899103657i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6538322312 - 0.7899103657i\) |
\(L(1)\) |
\(\approx\) |
\(0.8935952148 - 0.06717361359i\) |
\(L(1)\) |
\(\approx\) |
\(0.8935952148 - 0.06717361359i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.406 + 0.913i)T \) |
| 3 | \( 1 + (0.743 - 0.669i)T \) |
| 11 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (-0.587 + 0.809i)T \) |
| 17 | \( 1 + (0.207 - 0.978i)T \) |
| 19 | \( 1 + (-0.669 + 0.743i)T \) |
| 23 | \( 1 + (0.406 - 0.913i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.978 - 0.207i)T \) |
| 37 | \( 1 + (-0.994 - 0.104i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.207 - 0.978i)T \) |
| 53 | \( 1 + (0.743 - 0.669i)T \) |
| 59 | \( 1 + (-0.913 + 0.406i)T \) |
| 61 | \( 1 + (0.913 + 0.406i)T \) |
| 67 | \( 1 + (0.207 - 0.978i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.994 + 0.104i)T \) |
| 79 | \( 1 + (0.978 - 0.207i)T \) |
| 83 | \( 1 + (0.951 - 0.309i)T \) |
| 89 | \( 1 + (-0.913 - 0.406i)T \) |
| 97 | \( 1 + (0.951 + 0.309i)T \) |
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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
−27.75919294016148995200452106334, −26.55961094470538668485980428872, −25.84579799195837014593698591730, −25.02195178169513791951061703819, −23.362572239865492991968070945908, −22.258551749820986827392096127698, −21.49819244695072775116217798623, −20.586337675769529755681269876881, −19.78180865645053446233054687973, −19.11228441922736092489971135927, −17.7004475130860413584342806718, −16.963595024625216600999143847975, −15.48209100612524456369216395910, −14.67700998159788770242745987845, −13.31196799693790195377422231388, −12.5651309937074872433339348116, −11.12338645298296823395965144497, −10.173696258988092890314325939093, −9.445187579635081110726880129792, −8.3202801883246645327607954391, −7.38501640003638303203419848618, −5.129239135743421744805725155517, −4.07320172606015371175246145465, −2.91551893787258639907955556794, −1.78741071375725809233536895305,
0.37169793690431145388277935216, 1.94656533015973018280227519375, 3.64647132906890061909908103654, 5.24385664241092621124835092991, 6.57461332113919253133329168393, 7.35543072375411237871813169022, 8.53375789126583170071192436191, 9.16912229365828419400603449602, 10.51160645234708295097132820876, 12.091844429304983435169544792808, 13.358685953522978911367682707271, 14.198808104524216228891313183162, 14.891581917294405153217160659, 16.23067279335771755016236340730, 17.00690601098902644834300747827, 18.44130358402204296601266677370, 18.793523645840140874148627292669, 19.81666992716180886009706346310, 21.00991119881428401197590576162, 22.34051401711847255452435498630, 23.563929770698139546008996610157, 24.23649262779876432042432808317, 25.00890592977087929632138184548, 25.89043671622410890419754834426, 26.75652258058110990818388365330