| L(s) = 1 | + (0.642 − 0.766i)5-s + (0.5 − 0.866i)7-s − i·11-s + (0.642 + 0.766i)13-s + (0.766 + 0.642i)17-s + (0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + 31-s + (−0.342 − 0.939i)35-s − i·37-s + (−0.173 + 0.984i)41-s + (0.342 + 0.939i)43-s + (−0.939 − 0.342i)47-s + (−0.5 − 0.866i)49-s + ⋯ |
| L(s) = 1 | + (0.642 − 0.766i)5-s + (0.5 − 0.866i)7-s − i·11-s + (0.642 + 0.766i)13-s + (0.766 + 0.642i)17-s + (0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + 31-s + (−0.342 − 0.939i)35-s − i·37-s + (−0.173 + 0.984i)41-s + (0.342 + 0.939i)43-s + (−0.939 − 0.342i)47-s + (−0.5 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.086118204 - 1.197948421i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.086118204 - 1.197948421i\) |
| \(L(1)\) |
\(\approx\) |
\(1.368590468 - 0.3752462295i\) |
| \(L(1)\) |
\(\approx\) |
\(1.368590468 - 0.3752462295i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + (0.642 - 0.766i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (0.642 + 0.766i)T \) |
| 17 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.342 - 0.939i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.173 + 0.984i)T \) |
| 43 | \( 1 + (0.342 + 0.939i)T \) |
| 47 | \( 1 + (-0.939 - 0.342i)T \) |
| 53 | \( 1 + (0.984 - 0.173i)T \) |
| 59 | \( 1 + (0.342 + 0.939i)T \) |
| 61 | \( 1 + (0.642 + 0.766i)T \) |
| 67 | \( 1 + (-0.984 + 0.173i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 73 | \( 1 + (0.939 - 0.342i)T \) |
| 79 | \( 1 + (0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.939 + 0.342i)T \) |
| 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
−19.08252704648584531079355473089, −18.64138607826633331089806319520, −17.90218093526008381654686261651, −17.56805687596656222648441547718, −16.64567844839193226333705312475, −15.558878587957681266542344218516, −15.187493185691314865687011433515, −14.46418347113481336104288186573, −13.830260427814212075739550332985, −12.94605618128042981499128437292, −12.24722006352010504891542905773, −11.52845872545544861771228820653, −10.66329241842002802609624211769, −10.1148158982726744752720467430, −9.31622589128528553427913905058, −8.56134523793900730294670098733, −7.7082497986232220858603396130, −6.9082637755168942732729353158, −6.205978607155237327888279375507, −5.27634165933669349365905801364, −4.87930446355461496291201733200, −3.47775965566287105095600917376, −2.8050246424226075530923597527, −2.043386788657139290338696610480, −1.11931754737336764474946645284,
0.949165145466063610315168840091, 1.31118841183259206069686961269, 2.47684506746441544582443058414, 3.598806066050976005922562021, 4.29001079832669355179060749475, 5.101579549432459566545430067922, 5.940358417395375992881821847900, 6.53830995919919104833813268944, 7.63209913655218930490229860053, 8.35363903189471444105606531429, 8.90933438440521693767861692837, 9.84058160341398924614690137253, 10.46299722052245824520637706536, 11.34413818099958769986338067603, 11.85466049037318292011488094595, 13.08061248693973410991979199296, 13.36352156580945962701438008860, 14.11117047100486612851174833893, 14.69151082563446369410830982472, 15.81819868689933590203960219955, 16.5818270161498165286837586651, 16.86251802197424915335342080396, 17.65148586819387223148185603021, 18.340873472975001322486671392071, 19.36389961824997563805717810290
