L(s) = 1 | + (0.793 − 0.608i)2-s + (0.991 + 0.130i)3-s + (0.258 − 0.965i)4-s + (0.751 + 0.659i)5-s + (0.866 − 0.5i)6-s + (−0.946 − 0.321i)7-s + (−0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (0.997 + 0.0654i)10-s + (0.608 − 0.793i)11-s + (0.382 − 0.923i)12-s + (0.659 − 0.751i)13-s + (−0.946 + 0.321i)14-s + (0.659 + 0.751i)15-s + (−0.866 − 0.5i)16-s + (0.321 + 0.946i)17-s + ⋯ |
L(s) = 1 | + (0.793 − 0.608i)2-s + (0.991 + 0.130i)3-s + (0.258 − 0.965i)4-s + (0.751 + 0.659i)5-s + (0.866 − 0.5i)6-s + (−0.946 − 0.321i)7-s + (−0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (0.997 + 0.0654i)10-s + (0.608 − 0.793i)11-s + (0.382 − 0.923i)12-s + (0.659 − 0.751i)13-s + (−0.946 + 0.321i)14-s + (0.659 + 0.751i)15-s + (−0.866 − 0.5i)16-s + (0.321 + 0.946i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.454209077 - 1.753984969i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.454209077 - 1.753984969i\) |
\(L(1)\) |
\(\approx\) |
\(2.228674354 - 0.7756777578i\) |
\(L(1)\) |
\(\approx\) |
\(2.228674354 - 0.7756777578i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (0.793 - 0.608i)T \) |
| 3 | \( 1 + (0.991 + 0.130i)T \) |
| 5 | \( 1 + (0.751 + 0.659i)T \) |
| 7 | \( 1 + (-0.946 - 0.321i)T \) |
| 11 | \( 1 + (0.608 - 0.793i)T \) |
| 13 | \( 1 + (0.659 - 0.751i)T \) |
| 17 | \( 1 + (0.321 + 0.946i)T \) |
| 19 | \( 1 + (-0.195 + 0.980i)T \) |
| 23 | \( 1 + (0.442 - 0.896i)T \) |
| 29 | \( 1 + (-0.997 + 0.0654i)T \) |
| 31 | \( 1 + (-0.130 + 0.991i)T \) |
| 37 | \( 1 + (-0.896 + 0.442i)T \) |
| 41 | \( 1 + (-0.0654 - 0.997i)T \) |
| 43 | \( 1 + (-0.965 + 0.258i)T \) |
| 47 | \( 1 + (-0.707 - 0.707i)T \) |
| 53 | \( 1 + (0.608 + 0.793i)T \) |
| 59 | \( 1 + (-0.442 - 0.896i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.980 - 0.195i)T \) |
| 71 | \( 1 + (-0.0654 + 0.997i)T \) |
| 73 | \( 1 + (0.258 + 0.965i)T \) |
| 79 | \( 1 + (-0.923 + 0.382i)T \) |
| 83 | \( 1 + (0.946 - 0.321i)T \) |
| 89 | \( 1 + (0.382 + 0.923i)T \) |
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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
−30.17846460328928066815875957696, −29.30541520581374512252747028584, −27.93151724756719563576255321583, −26.232161459238266357990093159206, −25.60692387688992112869542361683, −24.95828355694741876797513950888, −23.965645053912491523409012407141, −22.657483814075962521992038689799, −21.539556281114324486672968220318, −20.70565322762390437913351570209, −19.68455934325299440336157824985, −18.20657632956367251950646294831, −16.85193556330992154057775311750, −15.84993205454926373872115495532, −14.806482237961946108885528574067, −13.55898032488298724165522405024, −13.103360203376011190248554779447, −11.873284114013038249397389403004, −9.48821734175804028849711997376, −8.96681278554980027371166914766, −7.30783518254591754244175665655, −6.285445590080019663946810373279, −4.74899684320175157472876713468, −3.400908296560307154219529717843, −1.98989437248518768271366833645,
1.550610111891095577564044308432, 3.12179800773345495299914642274, 3.71222432583836681959863707596, 5.789456862248631579258293352931, 6.81189409604627957684400615254, 8.75756593049185130999077492560, 10.08122540406993215074379390588, 10.67192768333737630371232832048, 12.61019225696415648156129582271, 13.469072605800991019190329417120, 14.30354976679435714028534771062, 15.222717070487783072609116265137, 16.58438454900806044908171057240, 18.56862435499991041475565588848, 19.213684170104633485789497820702, 20.30997890676380236125310962693, 21.281347586747554960223958308282, 22.17222554927610601008401899511, 23.098149113403303078164679818159, 24.61394535308989145608049213235, 25.432903888938252471214269136114, 26.41991359158087471786825185023, 27.65591464156332900678654452836, 29.11017304722790836767631718262, 29.91052753042721794724015755288