L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)6-s + 7-s − 8-s + (−0.5 − 0.866i)9-s + 12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s − 18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)6-s + 7-s − 8-s + (−0.5 − 0.866i)9-s + 12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s − 18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.431976594 + 0.3094889788i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.431976594 + 0.3094889788i\) |
\(L(1)\) |
\(\approx\) |
\(1.141605017 - 0.1169816302i\) |
\(L(1)\) |
\(\approx\) |
\(1.141605017 - 0.1169816302i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−21.925081942240592764606675460529, −20.641477345985804436135381811903, −20.19717006086645698080853112160, −18.608699001699750355802303185590, −18.28971070781456144784361493755, −17.554053748805463863487688238379, −16.85648573854119715756318145726, −16.08292510129237296118101855638, −15.09994198470349651335060106894, −14.39382304800065461900753180791, −13.57099277024717063683771051733, −12.95101625843581927010360211970, −12.13220809653482344519621515702, −11.33201312756217586887430151919, −10.57376841213485289808353017856, −8.935015370462266225689099403999, −8.366607770332024051015284709, −7.415622276488379713682207364901, −6.9594739699743654026844790658, −5.802115208654606052995227683832, −5.266045238489267989428660616171, −4.41781057662778852841296393754, −3.11563841566709222825185448114, −2.0157767325544476710325072021, −0.617193512313434169591465638981,
1.19875604478965166483486360527, 2.14162807053687868534115424770, 3.47821548847252338097108318794, 4.19164489509656093070885559278, 4.872396919594017237958495606505, 5.73547921260261043681556962285, 6.53921973853462991064287646876, 8.07766208133148031973569876044, 9.06390661006571470250096320658, 9.66758982879013797074991578361, 10.731561898835989103825520275258, 11.291798479003068134807979875032, 11.67866379456754566345971563568, 12.8247221332148667613156782464, 13.65445632207657991439048322021, 14.689748572163352634704510849370, 14.99470777264187811253301685699, 16.01080578092229760102376873976, 17.02077788053780794119698632135, 17.74872394656718152047203145849, 18.48805707079205358081731582099, 19.46529547396300598755795467206, 20.318156879861495046701550580884, 21.015461737672645216998354049403, 21.53817457451771688673245523834