L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s + i·7-s − i·8-s + (0.5 + 0.866i)9-s + 11-s + i·12-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s − i·18-s + (−0.5 + 0.866i)21-s + (−0.866 − 0.5i)22-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + (−0.5 − 0.866i)6-s + i·7-s − i·8-s + (0.5 + 0.866i)9-s + 11-s + i·12-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s − i·18-s + (−0.5 + 0.866i)21-s + (−0.866 − 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.041142134 + 0.8662451690i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.041142134 + 0.8662451690i\) |
\(L(1)\) |
\(\approx\) |
\(0.9402354877 + 0.2417258547i\) |
\(L(1)\) |
\(\approx\) |
\(0.9402354877 + 0.2417258547i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.866 + 0.5i)T \) |
| 47 | \( 1 + (0.866 - 0.5i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−29.81270295766736064298987947253, −28.64851063809106547101404904710, −27.16522652859113344865013567497, −26.63213462443873932691340912741, −25.601850049418624949271981930589, −24.58251903097824974840998503895, −23.97068354071482142469619624117, −22.56517298793284200345882755468, −20.72133975943750661938077010459, −19.761162556551667736405529793012, −19.307480924138913961738699983594, −17.77633500486283755356536970929, −17.124383419330932018089211943832, −15.6542587414641576078573446018, −14.54547811418445544737771276401, −13.73232895194969039580621526994, −12.1438084855807381392769913486, −10.52154366521334479934888586588, −9.469699581590003278390308447550, −8.299856402782605917204502996980, −7.29626487198053056254298344372, −6.342377504800982694730409299295, −4.17080147034489981236778821093, −2.247274541621864851946992312898, −0.7313878049530479436133578229,
1.87618786572459044516312858117, 3.00333292326442203381306374836, 4.5136508696141008441956911837, 6.71843632266816918758461571771, 8.16758565937952582535001923971, 9.149564978000021479758756351065, 9.82485564408653036957333150029, 11.38208595758594926299295341485, 12.38029009299466137667697536292, 13.9466327459888202958469207020, 15.23228015382199225222368757101, 16.176196723131572332557185405555, 17.44471568210826989506990840276, 18.72084056031961689893179480870, 19.58078175490721987160648978433, 20.37920336945404285303658969712, 21.76575071284274352988220566082, 22.04602215395298411164033839859, 24.51284653726250089246289663418, 25.12884932057098035336453661534, 26.17941230941762434807912187677, 27.13897975495466723384652711777, 27.86734918650052519788617299208, 28.98263183830920296346751707308, 30.16736371316128081691000398922