L(s) = 1 | + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (−0.984 + 0.173i)5-s + (0.173 + 0.984i)7-s + (−0.866 − 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.642 − 0.766i)13-s + (−0.866 + 0.5i)14-s + (0.173 − 0.984i)16-s + (−0.642 + 0.766i)17-s + (−0.342 + 0.939i)19-s + (0.642 − 0.766i)20-s + (−0.984 − 0.173i)22-s + (0.866 − 0.5i)23-s + ⋯ |
L(s) = 1 | + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (−0.984 + 0.173i)5-s + (0.173 + 0.984i)7-s + (−0.866 − 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.642 − 0.766i)13-s + (−0.866 + 0.5i)14-s + (0.173 − 0.984i)16-s + (−0.642 + 0.766i)17-s + (−0.342 + 0.939i)19-s + (0.642 − 0.766i)20-s + (−0.984 − 0.173i)22-s + (0.866 − 0.5i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07682554197 + 0.7412758471i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07682554197 + 0.7412758471i\) |
\(L(1)\) |
\(\approx\) |
\(0.5933628935 + 0.6223606053i\) |
\(L(1)\) |
\(\approx\) |
\(0.5933628935 + 0.6223606053i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 5 | \( 1 + (-0.984 + 0.173i)T \) |
| 7 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.642 - 0.766i)T \) |
| 17 | \( 1 + (-0.642 + 0.766i)T \) |
| 19 | \( 1 + (-0.342 + 0.939i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.866 + 0.5i)T \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.173 + 0.984i)T \) |
| 59 | \( 1 + (0.984 + 0.173i)T \) |
| 61 | \( 1 + (0.642 + 0.766i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + (0.939 + 0.342i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.984 - 0.173i)T \) |
| 83 | \( 1 + (-0.766 - 0.642i)T \) |
| 89 | \( 1 + (-0.984 - 0.173i)T \) |
| 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
−29.1499262689220341628306573862, −28.065735221079921495574214920441, −26.85959949743377934582769682084, −26.61593185564101459667341822618, −24.35967870746863577811840781495, −23.695804713495403059625200797481, −22.856978664544275070878217155084, −21.6307373123368371909520716421, −20.64068027136650725359999588278, −19.63732161130319081897933033451, −19.040015320131027431147005763041, −17.62463222961047703908853588460, −16.333946039778221617030589269255, −15.05592736217847251855043981622, −13.82373708162957144572546028206, −12.97841967808999830322685605101, −11.48900682321890726974375572108, −11.07763636249581852393259322625, −9.596306949256131345168855728910, −8.313368767366230004411747572960, −6.91896418288239468684396239459, −4.94729016450240295177827133591, −4.085173205659116589470606303312, −2.72526555987331113088574397149, −0.660513364573150067194878580968,
2.79932319385551585272548004306, 4.33228461753248495721517249602, 5.413162666261942555182448215734, 6.85903619663836250656093670246, 7.95158208890600246856340553971, 8.832476320115198557782273387970, 10.51473941200531308036876776196, 12.27284117286484412847076494704, 12.655163746216930270874442357849, 14.52632125220714327992873494864, 15.212110348556247575902847072333, 15.91727250700928239281981444762, 17.33523767005828406021078259717, 18.29177055679938380678892173167, 19.370750805195214733243494830830, 20.81271151056017533553114007536, 22.06224298188578814143593802951, 22.881544730701616334568380022085, 23.783143416612335819537596387165, 24.835848627420621550476432958277, 25.63521367378578076103323801658, 26.8995826431742410209170985060, 27.55821207767544235044386852422, 28.70650523149992717239826493076, 30.45313217844554426758432503615