L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.977 − 0.212i)3-s + (0.841 − 0.540i)4-s + (−0.755 − 0.654i)5-s + (−0.877 + 0.479i)6-s + (0.0713 + 0.997i)7-s + (−0.654 + 0.755i)8-s + (0.909 − 0.415i)9-s + (0.909 + 0.415i)10-s + (0.654 + 0.755i)11-s + (0.707 − 0.707i)12-s + (0.212 + 0.977i)13-s + (−0.349 − 0.936i)14-s + (−0.877 − 0.479i)15-s + (0.415 − 0.909i)16-s + (0.281 − 0.959i)17-s + ⋯ |
L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.977 − 0.212i)3-s + (0.841 − 0.540i)4-s + (−0.755 − 0.654i)5-s + (−0.877 + 0.479i)6-s + (0.0713 + 0.997i)7-s + (−0.654 + 0.755i)8-s + (0.909 − 0.415i)9-s + (0.909 + 0.415i)10-s + (0.654 + 0.755i)11-s + (0.707 − 0.707i)12-s + (0.212 + 0.977i)13-s + (−0.349 − 0.936i)14-s + (−0.877 − 0.479i)15-s + (0.415 − 0.909i)16-s + (0.281 − 0.959i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.463274227 + 0.2063215591i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.463274227 + 0.2063215591i\) |
\(L(1)\) |
\(\approx\) |
\(1.006958888 + 0.07534174884i\) |
\(L(1)\) |
\(\approx\) |
\(1.006958888 + 0.07534174884i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 89 | \( 1 \) |
good | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.977 - 0.212i)T \) |
| 5 | \( 1 + (-0.755 - 0.654i)T \) |
| 7 | \( 1 + (0.0713 + 0.997i)T \) |
| 11 | \( 1 + (0.654 + 0.755i)T \) |
| 13 | \( 1 + (0.212 + 0.977i)T \) |
| 17 | \( 1 + (0.281 - 0.959i)T \) |
| 19 | \( 1 + (0.349 - 0.936i)T \) |
| 23 | \( 1 + (0.936 + 0.349i)T \) |
| 29 | \( 1 + (0.997 - 0.0713i)T \) |
| 31 | \( 1 + (0.936 - 0.349i)T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (-0.212 + 0.977i)T \) |
| 43 | \( 1 + (0.997 + 0.0713i)T \) |
| 47 | \( 1 + (0.540 + 0.841i)T \) |
| 53 | \( 1 + (-0.540 + 0.841i)T \) |
| 59 | \( 1 + (-0.977 - 0.212i)T \) |
| 61 | \( 1 + (-0.599 - 0.800i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
| 71 | \( 1 + (0.755 - 0.654i)T \) |
| 73 | \( 1 + (-0.415 + 0.909i)T \) |
| 79 | \( 1 + (-0.909 - 0.415i)T \) |
| 83 | \( 1 + (0.877 - 0.479i)T \) |
| 97 | \( 1 + (-0.654 + 0.755i)T \) |
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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
−30.30036554448486440434244990889, −29.35187845477251717554709818433, −27.52910599929166175109135059995, −27.10318747888169021911852923369, −26.30157025000275911155941706927, −25.29795534811825894605101597675, −24.18249892827262309703240002807, −22.68838732457186282719643881796, −21.27088619643389504425888892902, −20.267492322199271317457710509691, −19.43937530356600619961353855871, −18.76157268576386239867917454243, −17.27909998193359050328742468601, −16.09249692301899938012476958640, −15.05554555827140801433981767694, −13.89536594538516200103404338191, −12.32590632564934720443584106801, −10.77541687072907317454058483049, −10.206570879245265422859468773415, −8.54866112328874516746883186811, −7.83578155897151731194428023640, −6.68677206691838731983177152763, −3.84121165319741181818263934026, −3.08505011207761590279860184290, −1.07901156677222961731904334046,
1.25527687420074306229806838466, 2.75490425500137803117584890742, 4.71833037299613541454609878012, 6.73640656489362559323222281684, 7.78866455886759664931158549057, 9.04258229299447287247996890424, 9.35933708928821086101307688112, 11.495366134292542953550984244135, 12.36716865892044770118881828092, 14.14263805722736814838719779469, 15.35413400041364286377744763758, 15.938078127296062496331932510203, 17.44040143396958762632558782568, 18.72329948421491645975808644499, 19.405409956430313467687696504460, 20.35497771969129654626981963744, 21.3155637023914676400509438415, 23.28733469393395054920944698135, 24.53588887899364793600771664487, 24.98323774628302063453193957520, 26.08736532208530560479022039182, 27.16641652661064501617555701391, 28.00060391909207343833873542360, 28.95187899430909142292688502063, 30.468133639869359769652914956417