L(s) = 1 | + (−0.608 + 0.793i)2-s + (−0.130 − 0.991i)3-s + (−0.258 − 0.965i)4-s + (−0.997 − 0.0654i)5-s + (0.866 + 0.5i)6-s + (0.442 − 0.896i)7-s + (0.923 + 0.382i)8-s + (−0.965 + 0.258i)9-s + (0.659 − 0.751i)10-s + (0.793 − 0.608i)11-s + (−0.923 + 0.382i)12-s + (0.0654 − 0.997i)13-s + (0.442 + 0.896i)14-s + (0.0654 + 0.997i)15-s + (−0.866 + 0.5i)16-s + (−0.896 + 0.442i)17-s + ⋯ |
L(s) = 1 | + (−0.608 + 0.793i)2-s + (−0.130 − 0.991i)3-s + (−0.258 − 0.965i)4-s + (−0.997 − 0.0654i)5-s + (0.866 + 0.5i)6-s + (0.442 − 0.896i)7-s + (0.923 + 0.382i)8-s + (−0.965 + 0.258i)9-s + (0.659 − 0.751i)10-s + (0.793 − 0.608i)11-s + (−0.923 + 0.382i)12-s + (0.0654 − 0.997i)13-s + (0.442 + 0.896i)14-s + (0.0654 + 0.997i)15-s + (−0.866 + 0.5i)16-s + (−0.896 + 0.442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01944097427 - 0.1978996545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01944097427 - 0.1978996545i\) |
\(L(1)\) |
\(\approx\) |
\(0.5125350931 - 0.1099540687i\) |
\(L(1)\) |
\(\approx\) |
\(0.5125350931 - 0.1099540687i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (-0.608 + 0.793i)T \) |
| 3 | \( 1 + (-0.130 - 0.991i)T \) |
| 5 | \( 1 + (-0.997 - 0.0654i)T \) |
| 7 | \( 1 + (0.442 - 0.896i)T \) |
| 11 | \( 1 + (0.793 - 0.608i)T \) |
| 13 | \( 1 + (0.0654 - 0.997i)T \) |
| 17 | \( 1 + (-0.896 + 0.442i)T \) |
| 19 | \( 1 + (-0.555 + 0.831i)T \) |
| 23 | \( 1 + (-0.321 + 0.946i)T \) |
| 29 | \( 1 + (-0.659 - 0.751i)T \) |
| 31 | \( 1 + (-0.991 + 0.130i)T \) |
| 37 | \( 1 + (-0.946 + 0.321i)T \) |
| 41 | \( 1 + (-0.751 + 0.659i)T \) |
| 43 | \( 1 + (0.965 + 0.258i)T \) |
| 47 | \( 1 + (0.707 - 0.707i)T \) |
| 53 | \( 1 + (0.793 + 0.608i)T \) |
| 59 | \( 1 + (0.321 + 0.946i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.831 + 0.555i)T \) |
| 71 | \( 1 + (-0.751 - 0.659i)T \) |
| 73 | \( 1 + (-0.258 + 0.965i)T \) |
| 79 | \( 1 + (-0.382 + 0.923i)T \) |
| 83 | \( 1 + (-0.442 - 0.896i)T \) |
| 89 | \( 1 + (-0.923 - 0.382i)T \) |
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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
−30.637228411278953292101177908158, −28.96717944347403645687333861021, −28.016581434888282423457851666976, −27.57988363398567171087273240179, −26.57707174082698336426425101232, −25.65848527585400247352645951965, −24.088360893577843566891343004750, −22.4949425329935589399577488061, −21.98869290294290111654167223385, −20.76132467941228302904436715855, −19.9313640650682418567516159635, −18.87889046851405834693763745315, −17.67159904609281719434033470592, −16.51208479371843776569807371665, −15.51667052273256037649775952205, −14.40603388473113439660208425715, −12.381112925026424670897492738, −11.54230849698935785363281667610, −10.80274934932098680437202024428, −9.12988590411199572630257009973, −8.73991886615209780489772869020, −6.98353442524505070545942350958, −4.72841678993408657389295579165, −3.87512964883210565695693400782, −2.27308333556590212440179083438,
0.11708075389643105107298665305, 1.40327444431266427808833122441, 3.94338914200933510878046115978, 5.699722818861764551870934753326, 6.977881525332766902196996057693, 7.858147460338105461835347595220, 8.66022009493243882387032578454, 10.62038186543759833690992942576, 11.552447278496076127756518254973, 13.11892652966158188800579242136, 14.23335103850980723842162217193, 15.31312789352304501283119235595, 16.71813496845973575671311078194, 17.40700060220493683992341343704, 18.57516638575118946158596260398, 19.59638685032432628442081672229, 20.16680852809051029612110121246, 22.531094923575145958291108615741, 23.3942548315715317716743555342, 24.136958021217367058513534618075, 24.91673893495723829925323285334, 26.17413013236368988825331826987, 27.2723336390191564714317494446, 27.88586980144593559071171503492, 29.34871828453978576406699490485