L(s) = 1 | + (−0.222 + 0.974i)2-s + (−0.623 + 0.781i)3-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)5-s + (−0.623 − 0.781i)6-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)10-s + (0.222 − 0.974i)11-s + (0.900 − 0.433i)12-s + (0.222 − 0.974i)13-s + (0.222 + 0.974i)15-s + (0.623 + 0.781i)16-s + (0.900 − 0.433i)17-s + 18-s − 19-s + ⋯ |
L(s) = 1 | + (−0.222 + 0.974i)2-s + (−0.623 + 0.781i)3-s + (−0.900 − 0.433i)4-s + (0.623 − 0.781i)5-s + (−0.623 − 0.781i)6-s + (0.623 − 0.781i)8-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)10-s + (0.222 − 0.974i)11-s + (0.900 − 0.433i)12-s + (0.222 − 0.974i)13-s + (0.222 + 0.974i)15-s + (0.623 + 0.781i)16-s + (0.900 − 0.433i)17-s + 18-s − 19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8627836505 - 0.3493023875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8627836505 - 0.3493023875i\) |
\(L(1)\) |
\(\approx\) |
\(0.7659110896 + 0.1746609998i\) |
\(L(1)\) |
\(\approx\) |
\(0.7659110896 + 0.1746609998i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (-0.222 + 0.974i)T \) |
| 3 | \( 1 + (-0.623 + 0.781i)T \) |
| 5 | \( 1 + (0.623 - 0.781i)T \) |
| 11 | \( 1 + (0.222 - 0.974i)T \) |
| 13 | \( 1 + (0.222 - 0.974i)T \) |
| 17 | \( 1 + (0.900 - 0.433i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.900 - 0.433i)T \) |
| 29 | \( 1 + (0.900 - 0.433i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.900 + 0.433i)T \) |
| 43 | \( 1 + (0.623 + 0.781i)T \) |
| 47 | \( 1 + (0.222 - 0.974i)T \) |
| 53 | \( 1 + (0.900 + 0.433i)T \) |
| 59 | \( 1 + (0.623 + 0.781i)T \) |
| 61 | \( 1 + (-0.900 + 0.433i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (-0.222 - 0.974i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (-0.222 - 0.974i)T \) |
| 89 | \( 1 + (0.222 + 0.974i)T \) |
| 97 | \( 1 - T \) |
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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
−19.716785522507912710119314443712, −19.23596593151692987633786914208, −18.62304989565395410116607637333, −17.9619549453346647934957347918, −17.3188566758337289306023089347, −16.94288750772210427680870005718, −15.75419669019654976744715122065, −14.42205942772639116843571158876, −14.09753426974496421757430711175, −13.28653156593671486230356952572, −12.4019028398845736858588871064, −12.01752225237672259988253689116, −11.16015604947138041979639104662, −10.39738593049056136786751045856, −9.953006771447928504513635725402, −8.94819160013201183936941882553, −8.0234331432586542734775148726, −7.159792512783813477807331641884, −6.45071180650253008887897331714, −5.60889878658932975756513910117, −4.611564950947423478602470498442, −3.74481008260461623864629427508, −2.54210439077354211702416037899, −1.93071106802093637307258280965, −1.26743902758132974826673837762,
0.43732370872447354692530694631, 1.20063927843350555140285475585, 2.92272758431990048082274429217, 4.057585422386289079944084411293, 4.69867335607621435253301987507, 5.67799721800066898085326938055, 5.876899446194316879959352270424, 6.72992466291952971122504921784, 8.11678561697339786618421082865, 8.5365600387085255349653076915, 9.32316386957054369729308181224, 10.29548792493234828567448063890, 10.40632032100571122958545315362, 11.83306487537447477885277549352, 12.49939746036222787501088298832, 13.503357276134704476257924965582, 14.03856206608880156642111546867, 14.96305328381919549777331220700, 15.670945096639483642106041990155, 16.40292219813288766606771937548, 16.727276408650354029903238792488, 17.54244636074922064119719002760, 17.98005876269016492982972522755, 18.94614224159159240320085659324, 19.829422254380242674010078312102