L(s) = 1 | + (−0.692 + 0.721i)2-s + (−0.919 + 0.391i)3-s + (−0.0402 − 0.999i)4-s + (−0.428 − 0.903i)5-s + (0.354 − 0.935i)6-s + (0.748 + 0.663i)8-s + (0.692 − 0.721i)9-s + (0.948 + 0.316i)10-s + (−0.692 − 0.721i)11-s + (0.428 + 0.903i)12-s + (0.748 + 0.663i)15-s + (−0.996 + 0.0804i)16-s + (−0.200 + 0.979i)17-s + (0.0402 + 0.999i)18-s + (0.5 − 0.866i)19-s + (−0.885 + 0.464i)20-s + ⋯ |
L(s) = 1 | + (−0.692 + 0.721i)2-s + (−0.919 + 0.391i)3-s + (−0.0402 − 0.999i)4-s + (−0.428 − 0.903i)5-s + (0.354 − 0.935i)6-s + (0.748 + 0.663i)8-s + (0.692 − 0.721i)9-s + (0.948 + 0.316i)10-s + (−0.692 − 0.721i)11-s + (0.428 + 0.903i)12-s + (0.748 + 0.663i)15-s + (−0.996 + 0.0804i)16-s + (−0.200 + 0.979i)17-s + (0.0402 + 0.999i)18-s + (0.5 − 0.866i)19-s + (−0.885 + 0.464i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3102196197 + 0.2489469853i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3102196197 + 0.2489469853i\) |
\(L(1)\) |
\(\approx\) |
\(0.4502798986 + 0.1077948284i\) |
\(L(1)\) |
\(\approx\) |
\(0.4502798986 + 0.1077948284i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.692 + 0.721i)T \) |
| 3 | \( 1 + (-0.919 + 0.391i)T \) |
| 5 | \( 1 + (-0.428 - 0.903i)T \) |
| 11 | \( 1 + (-0.692 - 0.721i)T \) |
| 17 | \( 1 + (-0.200 + 0.979i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.970 - 0.239i)T \) |
| 31 | \( 1 + (-0.987 + 0.160i)T \) |
| 37 | \( 1 + (0.632 + 0.774i)T \) |
| 41 | \( 1 + (-0.120 + 0.992i)T \) |
| 43 | \( 1 + (-0.354 - 0.935i)T \) |
| 47 | \( 1 + (0.0402 - 0.999i)T \) |
| 53 | \( 1 + (-0.200 + 0.979i)T \) |
| 59 | \( 1 + (0.996 + 0.0804i)T \) |
| 61 | \( 1 + (-0.200 - 0.979i)T \) |
| 67 | \( 1 + (0.845 + 0.534i)T \) |
| 71 | \( 1 + (-0.120 + 0.992i)T \) |
| 73 | \( 1 + (-0.692 - 0.721i)T \) |
| 79 | \( 1 + (-0.0402 + 0.999i)T \) |
| 83 | \( 1 + (-0.120 - 0.992i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.568 - 0.822i)T \) |
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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
−20.89632260759968081859521140451, −20.26911129985835046951276571388, −19.29326503017943538219842471984, −18.53929321399724466327616257600, −18.150861020101497961519467012145, −17.62403314855712741858498993520, −16.33317319592103607354243023352, −16.1546758954716168449055078585, −14.92576092465129673962729007199, −13.880237340838469451196612052950, −12.857603005493874323345611776030, −12.29575420295032526007280784630, −11.45698759285514268060644836441, −10.92280138008602228593133746695, −10.17419190363387612254292801714, −9.49554063004753435715646082596, −8.08800980068635832822820557303, −7.43360287943891251789522896764, −6.92565753057047357735699894571, −5.76485427673631644166399796671, −4.638733454753749866613654001937, −3.70464807756182131876964062447, −2.55955517498930429183065859076, −1.82116454428960060858253760704, −0.364807692724847432861099654872,
0.704522544835982704590854161556, 1.7568647920276194760406776075, 3.59114841926823736951662401986, 4.59057013991145033842859783297, 5.422359758596553613943330885404, 5.87386704522300387026326583938, 7.005965859996200400042104263, 7.84815655082542393110873331521, 8.667838502537143428467170395510, 9.44841505980879060434173242171, 10.24604164370041571747474270184, 11.17961385561049390442752238699, 11.64360947516096444813852124877, 12.91628993443642858292149888912, 13.4749994671589713380309026380, 14.84966194950809572648595466960, 15.585963177230998365548475550172, 16.01630980918807018751671812014, 16.84838606742465494774401089968, 17.2466955051103911070859425624, 18.196615780444823657743521740930, 18.82726950411252566646129257700, 19.85440199568114720527112909670, 20.418295867340721308265443878933, 21.5434272679252813850725589010