L(s) = 1 | + 5-s − 5·7-s + 4·13-s + 3·17-s − 4·19-s + 23-s + 25-s − 29-s + 31-s − 5·35-s − 37-s − 11·41-s + 4·43-s − 6·47-s + 18·49-s − 53-s + 59-s + 6·61-s + 4·65-s − 9·67-s − 13·71-s + 16·79-s − 9·83-s + 3·85-s − 8·89-s − 20·91-s − 4·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.88·7-s + 1.10·13-s + 0.727·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.185·29-s + 0.179·31-s − 0.845·35-s − 0.164·37-s − 1.71·41-s + 0.609·43-s − 0.875·47-s + 18/7·49-s − 0.137·53-s + 0.130·59-s + 0.768·61-s + 0.496·65-s − 1.09·67-s − 1.54·71-s + 1.80·79-s − 0.987·83-s + 0.325·85-s − 0.847·89-s − 2.09·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.170445742689843919525024130356, −7.10465666403230573269950239012, −6.46242274235722765967546513171, −6.04796151295771356608418029419, −5.23646751594445167084198979535, −4.00882265466375549405905426584, −3.38752501291883657140233227163, −2.63403822559365724391011744065, −1.36504756673912934394558790590, 0,
1.36504756673912934394558790590, 2.63403822559365724391011744065, 3.38752501291883657140233227163, 4.00882265466375549405905426584, 5.23646751594445167084198979535, 6.04796151295771356608418029419, 6.46242274235722765967546513171, 7.10465666403230573269950239012, 8.170445742689843919525024130356