L(s) = 1 | + (0.450 + 0.892i)2-s + (−0.210 + 0.977i)3-s + (−0.594 + 0.803i)4-s + (−0.127 + 0.991i)5-s + (−0.967 + 0.251i)6-s + (0.942 − 0.333i)7-s + (−0.985 − 0.169i)8-s + (−0.911 − 0.411i)9-s + (−0.942 + 0.333i)10-s + (0.594 + 0.803i)11-s + (−0.660 − 0.750i)12-s + (−0.372 + 0.927i)13-s + (0.721 + 0.691i)14-s + (−0.942 − 0.333i)15-s + (−0.292 − 0.956i)16-s + (0.210 − 0.977i)17-s + ⋯ |
L(s) = 1 | + (0.450 + 0.892i)2-s + (−0.210 + 0.977i)3-s + (−0.594 + 0.803i)4-s + (−0.127 + 0.991i)5-s + (−0.967 + 0.251i)6-s + (0.942 − 0.333i)7-s + (−0.985 − 0.169i)8-s + (−0.911 − 0.411i)9-s + (−0.942 + 0.333i)10-s + (0.594 + 0.803i)11-s + (−0.660 − 0.750i)12-s + (−0.372 + 0.927i)13-s + (0.721 + 0.691i)14-s + (−0.942 − 0.333i)15-s + (−0.292 − 0.956i)16-s + (0.210 − 0.977i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03137079976 + 1.187965711i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03137079976 + 1.187965711i\) |
\(L(1)\) |
\(\approx\) |
\(0.6180774206 + 0.9773351734i\) |
\(L(1)\) |
\(\approx\) |
\(0.6180774206 + 0.9773351734i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 149 | \( 1 \) |
good | 2 | \( 1 + (0.450 + 0.892i)T \) |
| 3 | \( 1 + (-0.210 + 0.977i)T \) |
| 5 | \( 1 + (-0.127 + 0.991i)T \) |
| 7 | \( 1 + (0.942 - 0.333i)T \) |
| 11 | \( 1 + (0.594 + 0.803i)T \) |
| 13 | \( 1 + (-0.372 + 0.927i)T \) |
| 17 | \( 1 + (0.210 - 0.977i)T \) |
| 19 | \( 1 + (0.0424 - 0.999i)T \) |
| 23 | \( 1 + (-0.372 - 0.927i)T \) |
| 29 | \( 1 + (0.873 + 0.487i)T \) |
| 31 | \( 1 + (0.372 + 0.927i)T \) |
| 37 | \( 1 + (-0.594 - 0.803i)T \) |
| 41 | \( 1 + (0.292 + 0.956i)T \) |
| 43 | \( 1 + (-0.524 + 0.851i)T \) |
| 47 | \( 1 + (0.0424 + 0.999i)T \) |
| 53 | \( 1 + (0.524 + 0.851i)T \) |
| 59 | \( 1 + (0.996 + 0.0848i)T \) |
| 61 | \( 1 + (-0.450 - 0.892i)T \) |
| 67 | \( 1 + (-0.721 + 0.691i)T \) |
| 71 | \( 1 + (0.127 - 0.991i)T \) |
| 73 | \( 1 + (0.778 + 0.628i)T \) |
| 79 | \( 1 + (-0.778 + 0.628i)T \) |
| 83 | \( 1 + (-0.778 - 0.628i)T \) |
| 89 | \( 1 + (0.828 - 0.559i)T \) |
| 97 | \( 1 + (-0.524 - 0.851i)T \) |
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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
−27.80466180103436738847303760249, −27.268092259774668489199285408227, −25.22398290483750022791348943270, −24.336172763165072005167681611085, −23.87757070922759288688078701941, −22.737357415318853108303064074802, −21.61304422712022129323031242576, −20.65906345333692114311774160838, −19.68915070606067288682882746916, −18.9348749404451188383113706545, −17.73358282648250637605690679255, −16.982838616866860675444592960093, −15.20992202148333822758337714108, −14.04719169054239948544975401744, −13.18899165396205245202699370859, −12.07713597954879843275814020198, −11.7099527471210261191297607036, −10.29573484687098996129071469379, −8.665726765513229511207430023037, −8.01513942305686569132519686250, −5.932560519205202499634033137125, −5.2600064839983705315509045444, −3.74055808007108679704651016161, −2.01913684124534306679707674367, −1.00650677858234921361134226977,
2.81589749334525567592075336662, 4.284218965268220213996391213457, 4.873940084610789135816329108722, 6.49371466462023630085342653054, 7.33249909189726360045601758206, 8.779473874965475875151905097242, 9.90989177063277179669078595718, 11.23986926046347855083895856788, 12.07437560350774920004995335419, 14.1902219313362371250284182471, 14.34103760712090645723230424128, 15.40526998013354993974002270071, 16.38605852071949156643327312344, 17.48847015324041240225372834930, 18.12095974697710087905228280790, 19.85143603426891568917656566271, 21.13604479372594675855458225141, 21.86583176091901372407786791896, 22.77363255429830247268226291664, 23.46152627490981909438820567019, 24.68998642240367361784190226405, 25.83750421841827151761910290719, 26.68065747555059790312698019262, 27.17744287011497222179624925752, 28.2688789284709024517088547998