L(s) = 1 | + (−0.399 + 0.916i)2-s + (−0.262 − 0.964i)3-s + (−0.681 − 0.732i)4-s + (−0.998 − 0.0483i)5-s + (0.989 + 0.144i)6-s + (0.168 − 0.985i)7-s + (0.943 − 0.331i)8-s + (−0.861 + 0.506i)9-s + (0.443 − 0.896i)10-s + (−0.527 + 0.849i)12-s + (0.443 + 0.896i)13-s + (0.836 + 0.548i)14-s + (0.215 + 0.976i)15-s + (−0.0724 + 0.997i)16-s + (−0.926 − 0.377i)17-s + (−0.120 − 0.992i)18-s + ⋯ |
L(s) = 1 | + (−0.399 + 0.916i)2-s + (−0.262 − 0.964i)3-s + (−0.681 − 0.732i)4-s + (−0.998 − 0.0483i)5-s + (0.989 + 0.144i)6-s + (0.168 − 0.985i)7-s + (0.943 − 0.331i)8-s + (−0.861 + 0.506i)9-s + (0.443 − 0.896i)10-s + (−0.527 + 0.849i)12-s + (0.443 + 0.896i)13-s + (0.836 + 0.548i)14-s + (0.215 + 0.976i)15-s + (−0.0724 + 0.997i)16-s + (−0.926 − 0.377i)17-s + (−0.120 − 0.992i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03297433880 - 0.4005784045i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03297433880 - 0.4005784045i\) |
\(L(1)\) |
\(\approx\) |
\(0.5865502017 - 0.06681251138i\) |
\(L(1)\) |
\(\approx\) |
\(0.5865502017 - 0.06681251138i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.399 + 0.916i)T \) |
| 3 | \( 1 + (-0.262 - 0.964i)T \) |
| 5 | \( 1 + (-0.998 - 0.0483i)T \) |
| 7 | \( 1 + (0.168 - 0.985i)T \) |
| 13 | \( 1 + (0.443 + 0.896i)T \) |
| 17 | \( 1 + (-0.926 - 0.377i)T \) |
| 19 | \( 1 + (0.970 + 0.239i)T \) |
| 23 | \( 1 + (0.0241 - 0.999i)T \) |
| 29 | \( 1 + (0.861 + 0.506i)T \) |
| 31 | \( 1 + (-0.607 - 0.794i)T \) |
| 37 | \( 1 + (-0.906 + 0.421i)T \) |
| 41 | \( 1 + (0.0724 + 0.997i)T \) |
| 43 | \( 1 + (-0.779 - 0.626i)T \) |
| 47 | \( 1 + (-0.748 - 0.663i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.644 - 0.764i)T \) |
| 61 | \( 1 + (0.809 + 0.587i)T \) |
| 67 | \( 1 + (0.779 - 0.626i)T \) |
| 71 | \( 1 + (0.568 - 0.822i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.354 - 0.935i)T \) |
| 83 | \( 1 + (-0.981 + 0.192i)T \) |
| 89 | \( 1 + (0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.715 - 0.698i)T \) |
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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
−20.95132902926268459382829191932, −19.97832987333570249345310466323, −19.70268873961553245141715654919, −18.712159392886192357312904144556, −17.787379611357342853416853834238, −17.501408576329007738699596534319, −16.07329556575246119626134912930, −15.82705457624762829987347743855, −15.05298316581758881030376086047, −14.09315169802184774640473776235, −12.95710952614160236443263531293, −12.19574158676059709682700752234, −11.45237554034640076078031226696, −11.090240944072220671647847275883, −10.18521705386720558962119931065, −9.35620418387233487375066030518, −8.54944313861399079351430845447, −8.14001646950006247630501761353, −6.88148778533495148444450557433, −5.460806197254426872673349069837, −4.93810991591973720709793895886, −3.82044372927010445283933937684, −3.325331221376048661685675914319, −2.4069227957030476376897942084, −0.97390279720965016900080019535,
0.14386565657946317948467825593, 0.849836503366151454528816108479, 1.83098840343122935842155404027, 3.4223754760673258431506495148, 4.453782397876922272204486718389, 5.10274754368139578728528578597, 6.456693217550895249144449522, 6.86791514777993688110301983724, 7.531040488966208843040573116759, 8.29752388942863473140784850883, 8.89680858041592284128868983179, 10.13528864563916663207574193107, 11.07448644519897396183242055858, 11.605016977222830216318411113675, 12.66943848426649443407685166147, 13.56997031545098137865833965542, 14.04192021475004311466088539722, 14.86242881739210959759564150127, 15.903208474969513929765874174136, 16.51951893390721209411935984273, 17.00538672055782919094612474841, 18.1078798030878557923273108845, 18.41758712051738772806685826143, 19.29719918869455372587311850002, 19.956713107478432747944073684828