L(s) = 1 | + (−0.755 − 0.654i)3-s + (−0.909 + 0.415i)7-s + (0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.909 − 0.415i)13-s + (−0.540 − 0.841i)17-s + (−0.841 − 0.540i)19-s + (0.959 + 0.281i)21-s + (0.540 − 0.841i)27-s + (−0.841 + 0.540i)29-s + (0.654 + 0.755i)31-s + (0.909 + 0.415i)33-s + (−0.989 + 0.142i)37-s + (0.415 + 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯ |
L(s) = 1 | + (−0.755 − 0.654i)3-s + (−0.909 + 0.415i)7-s + (0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.909 − 0.415i)13-s + (−0.540 − 0.841i)17-s + (−0.841 − 0.540i)19-s + (0.959 + 0.281i)21-s + (0.540 − 0.841i)27-s + (−0.841 + 0.540i)29-s + (0.654 + 0.755i)31-s + (0.909 + 0.415i)33-s + (−0.989 + 0.142i)37-s + (0.415 + 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5287867516 - 0.08450249616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5287867516 - 0.08450249616i\) |
\(L(1)\) |
\(\approx\) |
\(0.5650256763 - 0.08410578287i\) |
\(L(1)\) |
\(\approx\) |
\(0.5650256763 - 0.08410578287i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.755 - 0.654i)T \) |
| 7 | \( 1 + (-0.909 + 0.415i)T \) |
| 11 | \( 1 + (-0.959 + 0.281i)T \) |
| 13 | \( 1 + (-0.909 - 0.415i)T \) |
| 17 | \( 1 + (-0.540 - 0.841i)T \) |
| 19 | \( 1 + (-0.841 - 0.540i)T \) |
| 29 | \( 1 + (-0.841 + 0.540i)T \) |
| 31 | \( 1 + (0.654 + 0.755i)T \) |
| 37 | \( 1 + (-0.989 + 0.142i)T \) |
| 41 | \( 1 + (-0.142 + 0.989i)T \) |
| 43 | \( 1 + (0.755 + 0.654i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.909 + 0.415i)T \) |
| 59 | \( 1 + (0.415 - 0.909i)T \) |
| 61 | \( 1 + (0.654 + 0.755i)T \) |
| 67 | \( 1 + (0.281 - 0.959i)T \) |
| 71 | \( 1 + (0.959 + 0.281i)T \) |
| 73 | \( 1 + (0.540 - 0.841i)T \) |
| 79 | \( 1 + (-0.415 + 0.909i)T \) |
| 83 | \( 1 + (-0.989 + 0.142i)T \) |
| 89 | \( 1 + (-0.654 + 0.755i)T \) |
| 97 | \( 1 + (0.989 + 0.142i)T \) |
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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
−23.730725474266399870610427469299, −22.72882084436578320615653308781, −22.21092227769912466340236429496, −21.239124775889033260631837471369, −20.57465929847902210341931990447, −19.34147806055090620053645199827, −18.74129898208668678794828235827, −17.30011530073545382773593421328, −17.03109136087234643370244136299, −15.930994571138959271500805135111, −15.38307871172184794585413759715, −14.271132039209651973659295283528, −13.025583268663830986478133760630, −12.438784950321474327994375364267, −11.2598017852481615173496567973, −10.39914830004377058049157666149, −9.829792537062959539832147656375, −8.73973083971228158476472241714, −7.41910860993058077323790515767, −6.396702236570297961095085714537, −5.60538624589767070980832229851, −4.4340873959368777468477300125, −3.62667333325846968894144552465, −2.25670846741678300159555736114, −0.37778841836204229250805652233,
0.40666184054845363336632771496, 2.12498093804797364890140616293, 2.957837406932802969321784938909, 4.73619596198388174652319825592, 5.43564459761670297898155791515, 6.58905964689536616372676891530, 7.22653502342542757829689992449, 8.33699975368634320366516850531, 9.58028316417547898681729049174, 10.46142646165497404525064816401, 11.39958124542845940777030199446, 12.52317195395554522895443829537, 12.85611367510472139262608597108, 13.837775667497432819330244692235, 15.22397694010893934187686039616, 15.92423482921375616460217707498, 16.85106606184187539773840567410, 17.73961068819448689865749296656, 18.44898800083410683664885058933, 19.298316106716458737215417456997, 20.029436525926115259685175865691, 21.30710063218843905435736805911, 22.21345379178617159051256913474, 22.78661037886400784537340762439, 23.60531777438910702233932903855