L(s) = 1 | + (−0.691 + 0.722i)2-s + (0.936 − 0.351i)3-s + (−0.0448 − 0.998i)4-s + (−0.809 + 0.587i)5-s + (−0.393 + 0.919i)6-s + (0.134 + 0.990i)7-s + (0.753 + 0.657i)8-s + (0.753 − 0.657i)9-s + (0.134 − 0.990i)10-s + (0.858 + 0.512i)11-s + (−0.393 − 0.919i)12-s + (0.858 − 0.512i)13-s + (−0.809 − 0.587i)14-s + (−0.550 + 0.834i)15-s + (−0.995 + 0.0896i)16-s + (0.309 + 0.951i)17-s + ⋯ |
L(s) = 1 | + (−0.691 + 0.722i)2-s + (0.936 − 0.351i)3-s + (−0.0448 − 0.998i)4-s + (−0.809 + 0.587i)5-s + (−0.393 + 0.919i)6-s + (0.134 + 0.990i)7-s + (0.753 + 0.657i)8-s + (0.753 − 0.657i)9-s + (0.134 − 0.990i)10-s + (0.858 + 0.512i)11-s + (−0.393 − 0.919i)12-s + (0.858 − 0.512i)13-s + (−0.809 − 0.587i)14-s + (−0.550 + 0.834i)15-s + (−0.995 + 0.0896i)16-s + (0.309 + 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7435696411 + 0.3923502619i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7435696411 + 0.3923502619i\) |
\(L(1)\) |
\(\approx\) |
\(0.8653195336 + 0.3033580507i\) |
\(L(1)\) |
\(\approx\) |
\(0.8653195336 + 0.3033580507i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 \) |
good | 2 | \( 1 + (-0.691 + 0.722i)T \) |
| 3 | \( 1 + (0.936 - 0.351i)T \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 + (0.134 + 0.990i)T \) |
| 11 | \( 1 + (0.858 + 0.512i)T \) |
| 13 | \( 1 + (0.858 - 0.512i)T \) |
| 17 | \( 1 + (0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.550 - 0.834i)T \) |
| 23 | \( 1 + (-0.900 + 0.433i)T \) |
| 29 | \( 1 + (-0.963 - 0.266i)T \) |
| 31 | \( 1 + (-0.995 - 0.0896i)T \) |
| 37 | \( 1 + (-0.900 - 0.433i)T \) |
| 41 | \( 1 + (-0.222 - 0.974i)T \) |
| 43 | \( 1 + (0.983 - 0.178i)T \) |
| 47 | \( 1 + (0.936 + 0.351i)T \) |
| 53 | \( 1 + (-0.0448 + 0.998i)T \) |
| 59 | \( 1 + (-0.393 - 0.919i)T \) |
| 61 | \( 1 + (0.134 - 0.990i)T \) |
| 67 | \( 1 + (-0.0448 - 0.998i)T \) |
| 73 | \( 1 + (-0.691 + 0.722i)T \) |
| 79 | \( 1 + (0.753 + 0.657i)T \) |
| 83 | \( 1 + (-0.393 - 0.919i)T \) |
| 89 | \( 1 + (-0.0448 + 0.998i)T \) |
| 97 | \( 1 + (-0.222 + 0.974i)T \) |
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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
−31.38422399394179844118942280767, −30.41743840390948270770719578074, −29.513206988083508507960619214, −27.88389001402128861679892692447, −27.28386178946145573913937153222, −26.42195648917250190006796784288, −25.3079311520078547603328632623, −24.01970187052678497840418921005, −22.50177226037095460768530215296, −20.98695017142319641021220713072, −20.382996740833004976736590917203, −19.51252589447461810755756706075, −18.552726721607091541617093759228, −16.6765156340261706222666668878, −16.18489241973703107258538167795, −14.32579250486108466273013086422, −13.213442610149010213595543302835, −11.77177473194322298367561706430, −10.60399113057798595067498395154, −9.23806302377847146106136002235, −8.33528511077324385418982479179, −7.26518466495952479939258362501, −4.20549046076342322814919260434, −3.56606479746210687609160681244, −1.438239156288128546541990668457,
1.959627665280522336140143222142, 3.85346175715369471527302115753, 6.05739804423780310667602983212, 7.31574182059903877522160890061, 8.36381752989068341255847740468, 9.27688208486270349922054018756, 10.88804475782143427151759480494, 12.43385956560453673616895573646, 14.157109359850604095346936266, 15.14232651225369582153939536962, 15.6554855782935165599026709111, 17.54870914959488309395610835792, 18.623288454841979647542897035354, 19.32841261460462131034570763870, 20.32543411904505744790512558081, 22.10843465052358290405047237268, 23.50577706229716927176448394644, 24.42800924735244848500527861202, 25.61895307361630269004875635463, 26.05838773338124043121675046294, 27.55955549789625085250962781285, 28.03790788201154277106147535154, 29.955578528457115796438307806726, 30.84213872052155063863297697241, 31.94366872753933050163757487036