L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.793 + 0.608i)3-s + i·4-s + (0.608 − 0.793i)5-s + (−0.130 − 0.991i)6-s + (0.608 + 0.793i)7-s + (0.707 − 0.707i)8-s + (0.258 + 0.965i)9-s + (−0.991 + 0.130i)10-s + (−0.923 − 0.382i)11-s + (−0.608 + 0.793i)12-s + (0.866 + 0.5i)13-s + (0.130 − 0.991i)14-s + (0.965 − 0.258i)15-s − 16-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.793 + 0.608i)3-s + i·4-s + (0.608 − 0.793i)5-s + (−0.130 − 0.991i)6-s + (0.608 + 0.793i)7-s + (0.707 − 0.707i)8-s + (0.258 + 0.965i)9-s + (−0.991 + 0.130i)10-s + (−0.923 − 0.382i)11-s + (−0.608 + 0.793i)12-s + (0.866 + 0.5i)13-s + (0.130 − 0.991i)14-s + (0.965 − 0.258i)15-s − 16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.566912255 + 0.1977912302i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.566912255 + 0.1977912302i\) |
\(L(1)\) |
\(\approx\) |
\(1.160433746 + 0.02668861197i\) |
\(L(1)\) |
\(\approx\) |
\(1.160433746 + 0.02668861197i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.793 + 0.608i)T \) |
| 5 | \( 1 + (0.608 - 0.793i)T \) |
| 7 | \( 1 + (0.608 + 0.793i)T \) |
| 11 | \( 1 + (-0.923 - 0.382i)T \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
| 19 | \( 1 + (-0.258 + 0.965i)T \) |
| 23 | \( 1 + (-0.130 - 0.991i)T \) |
| 29 | \( 1 + (0.991 + 0.130i)T \) |
| 31 | \( 1 + (0.793 + 0.608i)T \) |
| 37 | \( 1 + (0.793 + 0.608i)T \) |
| 41 | \( 1 + (-0.382 + 0.923i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.258 - 0.965i)T \) |
| 59 | \( 1 + (-0.707 + 0.707i)T \) |
| 61 | \( 1 + (-0.608 - 0.793i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.130 - 0.991i)T \) |
| 73 | \( 1 + (-0.991 - 0.130i)T \) |
| 79 | \( 1 + (0.793 - 0.608i)T \) |
| 83 | \( 1 + (-0.258 + 0.965i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.382 + 0.923i)T \) |
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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
−22.96473539599554848192150449607, −21.45288978862418284854861318071, −20.668640626073568254229785515869, −19.91533567100771208056359924142, −19.07088084940464647827024418273, −18.22504440395636629248391337629, −17.769762213551820479557449698717, −17.18677051646532824244871004269, −15.61604847241092616246213032735, −15.29030342221576261033140755924, −14.19808237932833102691192924948, −13.715320615647445826446605822073, −13.04171974939850277796451079851, −11.36516003010888664334562919396, −10.57038589048988080889032904350, −9.8632589048613257982709111616, −8.88064433994521436983474340396, −7.89022190529639017846560030944, −7.42990907882066892016417273785, −6.55769885858542089497292489465, −5.682903202053309211743258851546, −4.39902546906002872789878384096, −2.951019122783503595961351222870, −2.00737122536275898572392619481, −0.96238206639914401132265737025,
1.346883475423141167160938864120, 2.21731647968357332491168434479, 3.048343034464455792810768628307, 4.30582314296816074357349128145, 5.05380471882769913460062919446, 6.323330263463694975714142073365, 8.11765828913136428601541593793, 8.32817657460992534666077709354, 9.02507622258589876888081029774, 10.010720654684273650782348780118, 10.5968377227784910385441618234, 11.678579355979957701516216639308, 12.60192777510794309109445158481, 13.442337065271967565909169315562, 14.15923220366533957080903420937, 15.37317116336914015145225602008, 16.25761805343553171186463349777, 16.69589277566894729080720585659, 18.01769650930888561909710166895, 18.47970384957246386054901157256, 19.35299296833443995594750961965, 20.40100454455255987155334415147, 20.90205747779674534443190725617, 21.38356991087055210998226847951, 21.94015156387242899589437216029