L(s) = 1 | + (−0.353 + 0.935i)2-s + (−0.994 + 0.104i)3-s + (−0.749 − 0.662i)4-s + (0.254 − 0.967i)6-s + (0.884 − 0.466i)8-s + (0.978 − 0.207i)9-s + (0.814 + 0.580i)12-s + (0.113 + 0.993i)13-s + (0.123 + 0.992i)16-s + (−0.0760 − 0.997i)17-s + (−0.151 + 0.988i)18-s + (−0.761 − 0.647i)19-s + (0.189 − 0.981i)23-s + (−0.830 + 0.556i)24-s + (−0.969 − 0.244i)26-s + (−0.951 + 0.309i)27-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.935i)2-s + (−0.994 + 0.104i)3-s + (−0.749 − 0.662i)4-s + (0.254 − 0.967i)6-s + (0.884 − 0.466i)8-s + (0.978 − 0.207i)9-s + (0.814 + 0.580i)12-s + (0.113 + 0.993i)13-s + (0.123 + 0.992i)16-s + (−0.0760 − 0.997i)17-s + (−0.151 + 0.988i)18-s + (−0.761 − 0.647i)19-s + (0.189 − 0.981i)23-s + (−0.830 + 0.556i)24-s + (−0.969 − 0.244i)26-s + (−0.951 + 0.309i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2004255891 + 0.5365557950i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2004255891 + 0.5365557950i\) |
\(L(1)\) |
\(\approx\) |
\(0.5238120140 + 0.2579624745i\) |
\(L(1)\) |
\(\approx\) |
\(0.5238120140 + 0.2579624745i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.353 + 0.935i)T \) |
| 3 | \( 1 + (-0.994 + 0.104i)T \) |
| 13 | \( 1 + (0.113 + 0.993i)T \) |
| 17 | \( 1 + (-0.0760 - 0.997i)T \) |
| 19 | \( 1 + (-0.761 - 0.647i)T \) |
| 23 | \( 1 + (0.189 - 0.981i)T \) |
| 29 | \( 1 + (0.610 + 0.791i)T \) |
| 31 | \( 1 + (0.953 - 0.299i)T \) |
| 37 | \( 1 + (-0.508 + 0.861i)T \) |
| 41 | \( 1 + (0.998 - 0.0570i)T \) |
| 43 | \( 1 + (-0.989 - 0.142i)T \) |
| 47 | \( 1 + (0.151 + 0.988i)T \) |
| 53 | \( 1 + (-0.992 - 0.123i)T \) |
| 59 | \( 1 + (-0.548 + 0.836i)T \) |
| 61 | \( 1 + (-0.161 + 0.986i)T \) |
| 67 | \( 1 + (-0.458 - 0.888i)T \) |
| 71 | \( 1 + (-0.0285 - 0.999i)T \) |
| 73 | \( 1 + (-0.956 - 0.290i)T \) |
| 79 | \( 1 + (0.345 + 0.938i)T \) |
| 83 | \( 1 + (0.856 + 0.516i)T \) |
| 89 | \( 1 + (-0.723 - 0.690i)T \) |
| 97 | \( 1 + (-0.441 + 0.897i)T \) |
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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
−17.99477779560066665874198344017, −17.38752901941944033493215229492, −17.25000113228663835020983167334, −16.23827868629014685868963969570, −15.576358750145916981175535331893, −14.74111234185601611774994752347, −13.701292873343335564214295996995, −13.06436584471682067764037792462, −12.5150835951342061243908758024, −11.96178949783743898810371092272, −11.18833379791451165248262682567, −10.57492735214855043408730544319, −10.1439613094436091480161853592, −9.40666138551258958385918504117, −8.29803849786006701750007018981, −7.94726019454335533507198670639, −6.95379551958319438776771842436, −6.04509014101848549497657709768, −5.41671690401549878809174657698, −4.528657454918080554533266301470, −3.87536778857186296146182716378, −3.038668730041125605428655011649, −1.95429417574152338700352170027, −1.30976501218056281735201346937, −0.301965320533994019174755432973,
0.77735532235088482992097812786, 1.63851895159757777406392001012, 2.87129074657865533389746547867, 4.301923428905894975540743818073, 4.584737297501508483730279307499, 5.28938688902412033267120551419, 6.34945982686580878846041414178, 6.57909792616480874797074012189, 7.27464704125793757708107188658, 8.18853224011142935127972274254, 9.04610361431217572377277265815, 9.52479660434192623111377241901, 10.4541371593274388982001482183, 10.90941017158206039860921429513, 11.79884542322790860435143528676, 12.45960686622711263695632348488, 13.37690610006087105366697585703, 13.90898691309308208946411639378, 14.78544330944677180078781400302, 15.42780471268263410906494538935, 16.17469365259421890264352917548, 16.51702132923739261657802538426, 17.23149194500471792976760823327, 17.79894878875947728297929254003, 18.47040944499904641781080841635