| L(s) = 1 | + (0.984 − 0.173i)3-s + (−0.866 − 0.5i)7-s + (0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.342 − 0.939i)17-s + (−0.939 − 0.342i)21-s + (0.642 − 0.766i)23-s + (0.866 − 0.5i)27-s + (0.939 − 0.342i)29-s + (0.5 − 0.866i)31-s + (0.642 + 0.766i)33-s − i·37-s − 39-s + (−0.173 − 0.984i)41-s + ⋯ |
| L(s) = 1 | + (0.984 − 0.173i)3-s + (−0.866 − 0.5i)7-s + (0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.342 − 0.939i)17-s + (−0.939 − 0.342i)21-s + (0.642 − 0.766i)23-s + (0.866 − 0.5i)27-s + (0.939 − 0.342i)29-s + (0.5 − 0.866i)31-s + (0.642 + 0.766i)33-s − i·37-s − 39-s + (−0.173 − 0.984i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.707391250 - 0.7891048687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707391250 - 0.7891048687i\) |
| \(L(1)\) |
\(\approx\) |
\(1.365387019 - 0.2606391198i\) |
| \(L(1)\) |
\(\approx\) |
\(1.365387019 - 0.2606391198i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (0.984 - 0.173i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.984 - 0.173i)T \) |
| 17 | \( 1 + (0.342 - 0.939i)T \) |
| 23 | \( 1 + (0.642 - 0.766i)T \) |
| 29 | \( 1 + (0.939 - 0.342i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.642 + 0.766i)T \) |
| 47 | \( 1 + (-0.342 - 0.939i)T \) |
| 53 | \( 1 + (0.642 - 0.766i)T \) |
| 59 | \( 1 + (0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.342 - 0.939i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.984 - 0.173i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.342 + 0.939i)T \) |
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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
−22.23760294365039637491206466475, −21.62910080796862116617845645975, −21.109942070549255252062377259872, −19.70289021328343531586000173896, −19.48503037807927859188411663521, −18.88061714201814557953919828887, −17.69844766744494803256744208300, −16.656921492065540311893090647870, −15.9639328410182328379085273958, −15.111584498545641078404361633638, −14.39583243017335722291091529683, −13.60018202491063714757118629876, −12.705626824237458677829920859275, −12.02104192446084660593516237721, −10.7176718077541779185787547024, −9.86282463038976561714686676512, −9.09877237196255624137363027514, −8.45930332208025823345943470472, −7.38986759006058321816246854640, −6.49836193678197838795235669652, −5.4516453262065514207941763224, −4.235413847452786350883939606171, −3.27513606332793737063827490342, −2.63418777134192913398839051980, −1.32987700477728807267241349526,
0.85458386458687140817515351589, 2.30052214832615045472808023134, 3.01254071218297330102638009119, 4.09418778838614091953870922216, 4.92166208225579520099438842376, 6.536596320167783242796394372334, 7.09403608445824013570862150677, 7.90430627948123506659854887157, 9.03615566875532957271138283873, 9.7966975950608953081810373523, 10.24668863059223975969893956264, 11.82631669208403381208853651213, 12.53510305259271174180984116382, 13.33150343452099502616991755091, 14.12909569535971725559674305778, 14.8902675766640278454281689232, 15.636482654070217952353539105055, 16.62686445904138063935559070400, 17.414367292006148519084680912420, 18.4728760148509880576727593225, 19.25284587795709138130425443269, 19.89286180321896628847645970242, 20.49067836035152987511127151266, 21.32016231473094290764637832919, 22.53498844443815685836482165147
