L(s) = 1 | + (−0.625 − 0.780i)2-s + (0.839 − 0.543i)3-s + (−0.217 + 0.975i)4-s + (−0.882 − 0.470i)5-s + (−0.948 − 0.315i)6-s + (0.250 + 0.968i)7-s + (0.897 − 0.440i)8-s + (0.409 − 0.912i)9-s + (0.184 + 0.982i)10-s + (−0.830 + 0.557i)11-s + (0.347 + 0.937i)12-s + (0.440 − 0.897i)13-s + (0.598 − 0.801i)14-s + (−0.996 + 0.0843i)15-s + (−0.905 − 0.425i)16-s + (0.954 − 0.299i)17-s + ⋯ |
L(s) = 1 | + (−0.625 − 0.780i)2-s + (0.839 − 0.543i)3-s + (−0.217 + 0.975i)4-s + (−0.882 − 0.470i)5-s + (−0.948 − 0.315i)6-s + (0.250 + 0.968i)7-s + (0.897 − 0.440i)8-s + (0.409 − 0.912i)9-s + (0.184 + 0.982i)10-s + (−0.830 + 0.557i)11-s + (0.347 + 0.937i)12-s + (0.440 − 0.897i)13-s + (0.598 − 0.801i)14-s + (−0.996 + 0.0843i)15-s + (−0.905 − 0.425i)16-s + (0.954 − 0.299i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.286889052 - 0.1065396354i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.286889052 - 0.1065396354i\) |
\(L(1)\) |
\(\approx\) |
\(0.8239176691 - 0.3066579835i\) |
\(L(1)\) |
\(\approx\) |
\(0.8239176691 - 0.3066579835i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 373 | \( 1 \) |
good | 2 | \( 1 + (-0.625 - 0.780i)T \) |
| 3 | \( 1 + (0.839 - 0.543i)T \) |
| 5 | \( 1 + (-0.882 - 0.470i)T \) |
| 7 | \( 1 + (0.250 + 0.968i)T \) |
| 11 | \( 1 + (-0.830 + 0.557i)T \) |
| 13 | \( 1 + (0.440 - 0.897i)T \) |
| 17 | \( 1 + (0.954 - 0.299i)T \) |
| 19 | \( 1 + (-0.571 + 0.820i)T \) |
| 23 | \( 1 + (-0.998 - 0.0506i)T \) |
| 29 | \( 1 + (-0.972 + 0.234i)T \) |
| 31 | \( 1 + (-0.347 + 0.937i)T \) |
| 37 | \( 1 + (0.378 + 0.925i)T \) |
| 41 | \( 1 + (0.979 - 0.201i)T \) |
| 43 | \( 1 + (0.975 + 0.217i)T \) |
| 47 | \( 1 + (0.993 - 0.117i)T \) |
| 53 | \( 1 + (0.912 - 0.409i)T \) |
| 59 | \( 1 + (-0.283 + 0.959i)T \) |
| 61 | \( 1 + (0.543 + 0.839i)T \) |
| 67 | \( 1 + (0.848 - 0.528i)T \) |
| 71 | \( 1 + (-0.736 + 0.676i)T \) |
| 73 | \( 1 + (0.585 - 0.810i)T \) |
| 79 | \( 1 + (-0.982 + 0.184i)T \) |
| 83 | \( 1 + (0.409 + 0.912i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.299 + 0.954i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.27529465736011847429164887244, −23.783166788001363012811707543030, −23.01040393625877690523321724716, −21.75401072008400505582349300523, −20.68360896341624875880613389513, −19.8661758968787546118939432182, −19.03787715027860928431138501816, −18.51306216808911965987227202108, −17.10831040304966748546039235407, −16.213994920224885179000928346426, −15.70516314692318019640097304975, −14.63198369943219658088812814840, −14.119971712680076774180505558170, −13.143050036335834012761352927246, −11.18283127599665654181091460878, −10.70843090887530273081958326545, −9.69275062549052916402183592815, −8.619784749517152974618314199567, −7.75023093386334231837582664023, −7.30758561606846527487797415554, −5.87178754423140625397102960372, −4.41772927118633491660131067999, −3.76999391297983743261509538769, −2.193312247373511666111721795751, −0.494602038893392309555798692331,
0.97450244486841092061739257681, 2.15173294761073746655424980673, 3.13109965208816664914048691183, 4.11449319503481658994446535946, 5.571544962733591843374844706119, 7.403209143052972787173920907377, 8.02276562873146494974486868363, 8.61144298442046462189185502074, 9.630616105396794537517060180360, 10.711194366991896035804512930579, 12.060128582737530163963758225399, 12.414691954591031057512775988921, 13.20606752227705818310540516079, 14.59425491771484127921127941016, 15.517221941041049569977794275773, 16.34915484898456950814216517243, 17.79114333778976415616547717640, 18.457812402085026592368271891776, 19.040055331905581622379177143839, 20.0281805873728326574254237414, 20.66216834262302735064030529402, 21.24329768980661641781628766504, 22.627681589673353014255594045395, 23.51604597946284572663168292627, 24.57723420726888488725541382133