L(s) = 1 | + (0.836 + 0.548i)2-s + (0.0241 + 0.999i)3-s + (0.399 + 0.916i)4-s + (0.958 + 0.285i)5-s + (−0.527 + 0.849i)6-s + (−0.970 + 0.239i)7-s + (−0.168 + 0.985i)8-s + (−0.998 + 0.0483i)9-s + (0.644 + 0.764i)10-s + (−0.906 + 0.421i)12-s + (−0.970 + 0.239i)13-s + (−0.943 − 0.331i)14-s + (−0.262 + 0.964i)15-s + (−0.681 + 0.732i)16-s + (−0.906 − 0.421i)17-s + (−0.861 − 0.506i)18-s + ⋯ |
L(s) = 1 | + (0.836 + 0.548i)2-s + (0.0241 + 0.999i)3-s + (0.399 + 0.916i)4-s + (0.958 + 0.285i)5-s + (−0.527 + 0.849i)6-s + (−0.970 + 0.239i)7-s + (−0.168 + 0.985i)8-s + (−0.998 + 0.0483i)9-s + (0.644 + 0.764i)10-s + (−0.906 + 0.421i)12-s + (−0.970 + 0.239i)13-s + (−0.943 − 0.331i)14-s + (−0.262 + 0.964i)15-s + (−0.681 + 0.732i)16-s + (−0.906 − 0.421i)17-s + (−0.861 − 0.506i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.6836888514 + 1.628048854i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.6836888514 + 1.628048854i\) |
\(L(1)\) |
\(\approx\) |
\(0.8526483587 + 1.191828742i\) |
\(L(1)\) |
\(\approx\) |
\(0.8526483587 + 1.191828742i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (0.836 + 0.548i)T \) |
| 3 | \( 1 + (0.0241 + 0.999i)T \) |
| 5 | \( 1 + (0.958 + 0.285i)T \) |
| 7 | \( 1 + (-0.970 + 0.239i)T \) |
| 13 | \( 1 + (-0.970 + 0.239i)T \) |
| 17 | \( 1 + (-0.906 - 0.421i)T \) |
| 19 | \( 1 + (0.981 + 0.192i)T \) |
| 23 | \( 1 + (-0.443 + 0.896i)T \) |
| 29 | \( 1 + (-0.354 + 0.935i)T \) |
| 31 | \( 1 + (0.715 - 0.698i)T \) |
| 37 | \( 1 + (-0.748 + 0.663i)T \) |
| 41 | \( 1 + (0.485 - 0.873i)T \) |
| 43 | \( 1 + (0.958 + 0.285i)T \) |
| 47 | \( 1 + (-0.998 + 0.0483i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.906 - 0.421i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 + (0.958 - 0.285i)T \) |
| 71 | \( 1 + (-0.989 + 0.144i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.943 + 0.331i)T \) |
| 83 | \( 1 + (0.399 - 0.916i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (-0.0724 + 0.997i)T \) |
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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
−20.15445368130487678988024380652, −19.64098757354912255123327227882, −19.00723691046232577986250061981, −18.03120158431057808033068975898, −17.428274092897611565284971875097, −16.48957591510583819487674963003, −15.63104244537501400321147183731, −14.5129920954228122685203185153, −13.95458103552279933278029044633, −13.25839783972174731517334253694, −12.72508243083481714102780673436, −12.189528835156092037118703589486, −11.20643223711691854516998618064, −10.21102553304838293419429946976, −9.63155079130931713175056329038, −8.725481202266091410459594152157, −7.409589650811769095488035206666, −6.61932520322854362436381550655, −6.06639259795217494837629258670, −5.271743884457541850481855989587, −4.28313224188893288858850088117, −2.970722310200099140808299619903, −2.4787362967135551083326314926, −1.55156541813139032978364717240, −0.43838275079069941856308948108,
2.11722531072919104674118848577, 2.91776554129254057670570781356, 3.55893339070041566400008490596, 4.69974259189567853129493174567, 5.338071669424603164451155120037, 6.09377948871629250554474224473, 6.81591834511615750055040895833, 7.7730016608289049192922363939, 9.14174767503546239913218627202, 9.434718557726390455473599074149, 10.34100027522966670644506894252, 11.30545477726435839653876766040, 12.11103028296647069034079811975, 13.01551377881438784539164480157, 13.8679052273573899245847847452, 14.26483882655170584226024395195, 15.25021204676545673866681128792, 15.79970012789925263980498224547, 16.463768621565016487505324819856, 17.27023544705855174020752988033, 17.78217114800633626361510200490, 19.02188810583638641027234615151, 20.06477117841377727928238716177, 20.5985576861173505855344147432, 21.54501827208551600081732198041