L(s) = 1 | − 3-s + 4·7-s + 9-s − 4·11-s − 6·13-s + 6·17-s + 4·19-s − 4·21-s − 4·23-s − 27-s + 6·29-s + 8·31-s + 4·33-s − 2·37-s + 6·39-s + 2·41-s − 43-s + 4·47-s + 9·49-s − 6·51-s + 6·53-s − 4·57-s + 12·59-s + 10·61-s + 4·63-s + 12·67-s + 4·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-s − 0.872·21-s − 0.834·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s − 0.328·37-s + 0.960·39-s + 0.312·41-s − 0.152·43-s + 0.583·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s + 1.56·59-s + 1.28·61-s + 0.503·63-s + 1.46·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.407656568\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.407656568\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.43743606642060, −14.03227106955313, −13.64047545546885, −12.77897903962033, −12.23021886363757, −11.96123448680008, −11.56624133395658, −10.87217295344064, −10.28801674738317, −9.954849085812831, −9.586053974317821, −8.454053066969823, −8.066700448168479, −7.749929103273950, −7.188688292760989, −6.582371861671631, −5.577410142304655, −5.256367083440613, −5.003262436308138, −4.330560005418422, −3.541882587909196, −2.471632958195974, −2.344269698339962, −1.177254940475678, −0.6351378938478185,
0.6351378938478185, 1.177254940475678, 2.344269698339962, 2.471632958195974, 3.541882587909196, 4.330560005418422, 5.003262436308138, 5.256367083440613, 5.577410142304655, 6.582371861671631, 7.188688292760989, 7.749929103273950, 8.066700448168479, 8.454053066969823, 9.586053974317821, 9.954849085812831, 10.28801674738317, 10.87217295344064, 11.56624133395658, 11.96123448680008, 12.23021886363757, 12.77897903962033, 13.64047545546885, 14.03227106955313, 14.43743606642060