L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (0.923 + 0.382i)5-s − 6-s + (0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s − i·9-s + (0.382 + 0.923i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (−0.923 − 0.382i)13-s + (0.923 + 0.382i)14-s + (−0.923 + 0.382i)15-s − 16-s + (−0.923 − 0.382i)17-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (0.923 + 0.382i)5-s − 6-s + (0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s − i·9-s + (0.382 + 0.923i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (−0.923 − 0.382i)13-s + (0.923 + 0.382i)14-s + (−0.923 + 0.382i)15-s − 16-s + (−0.923 − 0.382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7195034467 + 1.090196433i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7195034467 + 1.090196433i\) |
\(L(1)\) |
\(\approx\) |
\(1.018368144 + 0.8371583131i\) |
\(L(1)\) |
\(\approx\) |
\(1.018368144 + 0.8371583131i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.923 + 0.382i)T \) |
| 7 | \( 1 + (0.923 - 0.382i)T \) |
| 11 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + (-0.923 - 0.382i)T \) |
| 17 | \( 1 + (-0.923 - 0.382i)T \) |
| 19 | \( 1 + (0.923 + 0.382i)T \) |
| 23 | \( 1 + (0.382 - 0.923i)T \) |
| 29 | \( 1 + (0.382 - 0.923i)T \) |
| 31 | \( 1 + (0.707 - 0.707i)T \) |
| 37 | \( 1 + (-0.382 - 0.923i)T \) |
| 41 | \( 1 + (-0.382 + 0.923i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.707 - 0.707i)T \) |
| 59 | \( 1 + (0.382 + 0.923i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.923 - 0.382i)T \) |
| 71 | \( 1 + (-0.382 - 0.923i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.707 - 0.707i)T \) |
| 83 | \( 1 + (0.923 + 0.382i)T \) |
| 89 | \( 1 + (-0.707 + 0.707i)T \) |
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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
−29.564796266244381009029956536627, −28.980861237292382987403711406802, −28.25127910528354103067603415378, −26.98016684680569563954355110944, −25.0631042817907373943988270597, −24.21579549095195024553292587952, −23.72743768575642131837322576264, −22.01106167052246786614851879134, −21.71814322319928272393461316171, −20.456318616819286740330109896104, −19.147085488686319935081795904244, −18.09128135776569827477593326422, −17.28300416571397953599215982583, −15.68395885387361440964850244528, −14.09427568294314108985710501487, −13.40579377537937880255371577283, −12.280228323891796535920251178687, −11.33962213424888842159407576014, −10.27399631768084377518061281693, −8.71562336282065583657317511106, −6.86879751724370292912608365322, −5.432966084220273174244268696726, −4.97545871934902319279881368531, −2.53071296480721811866723847296, −1.41679223254449677072988432355,
2.614512229051286648511670053163, 4.5378882672659743431410016728, 5.20914030715717297820385352924, 6.49262156995221291145048925454, 7.73119254724387419455558449843, 9.51780530270899800744507429658, 10.68287637843254956573041086150, 11.90777282512552211333410301380, 13.2288463446882322166052856871, 14.47003046637500504491345148376, 15.21332886593742392135639936240, 16.48468284714363968708170510861, 17.66320752700602947486089483433, 17.879509874591297242872622886690, 20.5859880596814999919045202855, 21.101808043423506605071212955009, 22.32015302673595627292643128891, 22.83104368270275699712865322250, 24.16658726965734530026954579927, 25.00862908132174696022558976000, 26.53571439731544595717508101066, 26.80353105001990405966987786478, 28.46114664147683213436409187617, 29.476377316893449692228653046388, 30.4888900310269153691159329209