L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s − 13-s + 15-s + 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s + 33-s − 35-s + 37-s − 39-s − 41-s − 43-s + 45-s − 47-s + 49-s + 51-s + 53-s + 55-s + ⋯ |
L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s − 13-s + 15-s + 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s + 33-s − 35-s + 37-s − 39-s − 41-s − 43-s + 45-s − 47-s + 49-s + 51-s + 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4504 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.400766299\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.400766299\) |
\(L(1)\) |
\(\approx\) |
\(1.805048248\) |
\(L(1)\) |
\(\approx\) |
\(1.805048248\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 563 | \( 1 \) |
good | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 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
−18.27352527897573050173521317318, −17.69518802563632274802841934591, −16.653578593575161356822336490479, −16.46262829996489997027588569296, −15.44203994706576255034405318083, −14.74202939255166302746212237285, −14.0898524711241264296859240340, −13.73138277566908257429993171225, −12.98064577195226758393649014052, −12.226402456957102478830518762, −11.74303162661843896182169563549, −10.14214509223651037978288034121, −9.97143362320599164052994951462, −9.55870920369413599854696786413, −8.73401728844849290068074651208, −7.990920411206225037320663474636, −7.05842757499228486760474394940, −6.581239914693895336361350778825, −5.77845338964333548617175807465, −4.87985734653841751692383034994, −3.99943090169576143200578481743, −3.11588813363248839591899878759, −2.70875017175042853521520837149, −1.73396719610938299413366072624, −0.9725770762554705406132363777,
0.9725770762554705406132363777, 1.73396719610938299413366072624, 2.70875017175042853521520837149, 3.11588813363248839591899878759, 3.99943090169576143200578481743, 4.87985734653841751692383034994, 5.77845338964333548617175807465, 6.581239914693895336361350778825, 7.05842757499228486760474394940, 7.990920411206225037320663474636, 8.73401728844849290068074651208, 9.55870920369413599854696786413, 9.97143362320599164052994951462, 10.14214509223651037978288034121, 11.74303162661843896182169563549, 12.226402456957102478830518762, 12.98064577195226758393649014052, 13.73138277566908257429993171225, 14.0898524711241264296859240340, 14.74202939255166302746212237285, 15.44203994706576255034405318083, 16.46262829996489997027588569296, 16.653578593575161356822336490479, 17.69518802563632274802841934591, 18.27352527897573050173521317318