L(s) = 1 | + (0.342 + 0.939i)2-s + (0.984 + 0.173i)3-s + (−0.766 + 0.642i)4-s + (0.173 + 0.984i)6-s + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.866 + 0.5i)12-s + (−0.984 + 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (0.342 + 0.939i)17-s + i·18-s + (−0.939 + 0.342i)21-s + (−0.984 − 0.173i)22-s + ⋯ |
L(s) = 1 | + (0.342 + 0.939i)2-s + (0.984 + 0.173i)3-s + (−0.766 + 0.642i)4-s + (0.173 + 0.984i)6-s + (−0.866 + 0.5i)7-s + (−0.866 − 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.5 + 0.866i)11-s + (−0.866 + 0.5i)12-s + (−0.984 + 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (0.342 + 0.939i)17-s + i·18-s + (−0.939 + 0.342i)21-s + (−0.984 − 0.173i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1767802170 + 1.868072433i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1767802170 + 1.868072433i\) |
\(L(1)\) |
\(\approx\) |
\(0.9223565861 + 0.9931757242i\) |
\(L(1)\) |
\(\approx\) |
\(0.9223565861 + 0.9931757242i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 3 | \( 1 + (0.984 + 0.173i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.984 + 0.173i)T \) |
| 17 | \( 1 + (0.342 + 0.939i)T \) |
| 23 | \( 1 + (0.642 + 0.766i)T \) |
| 29 | \( 1 + (0.939 + 0.342i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.642 + 0.766i)T \) |
| 47 | \( 1 + (-0.342 + 0.939i)T \) |
| 53 | \( 1 + (0.642 + 0.766i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.342 + 0.939i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.984 + 0.173i)T \) |
| 79 | \( 1 + (-0.173 + 0.984i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.173 - 0.984i)T \) |
| 97 | \( 1 + (0.342 + 0.939i)T \) |
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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
−29.48330036745819583981651496752, −28.90450713008216150070872378823, −27.07796898200707536148837042582, −26.70738251597767160050846473916, −25.26897192213336807253471232055, −24.1190196008869760289567351059, −23.018794871997052769039860065100, −21.85294329714583465939167076019, −20.84607498271861998889276216975, −19.86815435623128000085061756300, −19.16426091763895587643733981010, −18.198270858395307907851027038190, −16.4244020102644219776892076580, −15.007569792172347794277779056984, −13.88806909531061409519570712478, −13.165931551949986314580798249970, −12.08531733577547896342186510867, −10.43667545784786827476793935019, −9.61021203171702628485831432351, −8.375271067209906133229225270198, −6.80971678393313520431257633691, −4.948119818026468540083078003138, −3.41231115360505481604069947558, −2.61136917444070395285106627891, −0.66769473517189660564445617831,
2.55597237984191897337141684415, 3.89119442824375811129816938206, 5.27218632101250518940891513829, 6.86568591240963037166991393461, 7.86017759675165875025532754001, 9.1514200243059098579517436602, 9.97829340533415089742588057409, 12.44098821233523486340195659076, 13.087480482507805945294002787446, 14.46433164496284062734975852764, 15.22478334540377254566419336291, 16.10412777332303770928350032251, 17.418043975058359146609800650165, 18.76154287650181773569324831111, 19.71348438815784601881966168497, 21.19720271091912799640121382338, 22.01392563004468109425444409266, 23.19992730323075993973255197939, 24.37425853069745485940614474263, 25.407476203211366458521523823429, 25.92172520323100338441854735468, 26.91429010043483726294250617971, 28.09104119273540869789990167435, 29.62812808914594419659100819449, 30.98135770518217531849514316454