L(s) = 1 | + 3·5-s − 7-s − 6·11-s − 4·13-s + 19-s + 4·23-s + 4·25-s − 6·29-s − 3·35-s + 12·37-s + 6·41-s + 4·43-s − 9·47-s − 6·49-s − 5·53-s − 18·55-s − 61-s − 12·65-s + 10·67-s − 12·71-s + 11·73-s + 6·77-s + 12·79-s − 4·83-s + 14·89-s + 4·91-s + 3·95-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 0.377·7-s − 1.80·11-s − 1.10·13-s + 0.229·19-s + 0.834·23-s + 4/5·25-s − 1.11·29-s − 0.507·35-s + 1.97·37-s + 0.937·41-s + 0.609·43-s − 1.31·47-s − 6/7·49-s − 0.686·53-s − 2.42·55-s − 0.128·61-s − 1.48·65-s + 1.22·67-s − 1.42·71-s + 1.28·73-s + 0.683·77-s + 1.35·79-s − 0.439·83-s + 1.48·89-s + 0.419·91-s + 0.307·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.723563813\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.723563813\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 463 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−13.03921714900835, −12.83097678962425, −12.36231708956307, −11.56929786134858, −11.00540446027972, −10.78008460476670, −10.02820562362030, −9.840724586384125, −9.360055476706783, −9.096878581002175, −8.159370931385820, −7.703774568659638, −7.521114068600047, −6.642661082597287, −6.336824926019954, −5.632637408467380, −5.349381723853679, −4.889808014329080, −4.371047285465797, −3.415937273444022, −2.874481920319444, −2.403100287657439, −2.080466501111067, −1.191597768844090, −0.3666844736438064,
0.3666844736438064, 1.191597768844090, 2.080466501111067, 2.403100287657439, 2.874481920319444, 3.415937273444022, 4.371047285465797, 4.889808014329080, 5.349381723853679, 5.632637408467380, 6.336824926019954, 6.642661082597287, 7.521114068600047, 7.703774568659638, 8.159370931385820, 9.096878581002175, 9.360055476706783, 9.840724586384125, 10.02820562362030, 10.78008460476670, 11.00540446027972, 11.56929786134858, 12.36231708956307, 12.83097678962425, 13.03921714900835