| L(s) = 1 | + 2·3-s + 2·7-s + 9-s − 4·11-s + 4·13-s − 4·19-s + 4·21-s − 2·23-s − 4·27-s + 2·29-s − 8·33-s + 4·37-s + 8·39-s + 2·41-s − 6·43-s − 6·47-s − 3·49-s − 4·53-s − 8·57-s − 12·59-s − 10·61-s + 2·63-s + 14·67-s − 4·69-s + 8·71-s + 8·73-s − 8·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s − 0.917·19-s + 0.872·21-s − 0.417·23-s − 0.769·27-s + 0.371·29-s − 1.39·33-s + 0.657·37-s + 1.28·39-s + 0.312·41-s − 0.914·43-s − 0.875·47-s − 3/7·49-s − 0.549·53-s − 1.05·57-s − 1.56·59-s − 1.28·61-s + 0.251·63-s + 1.71·67-s − 0.481·69-s + 0.949·71-s + 0.936·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.660108418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.660108418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−12.73146010216774514797104134533, −11.33359133996889118176969030818, −10.53622600687691340454258542756, −9.301178546757553526958267434221, −8.212252834472441096002849564389, −7.930378831312213533244284917039, −6.25352345843664090435411695863, −4.82604799709023040346819665152, −3.41809103272636300318248682927, −2.08793608524985768454862433122,
2.08793608524985768454862433122, 3.41809103272636300318248682927, 4.82604799709023040346819665152, 6.25352345843664090435411695863, 7.930378831312213533244284917039, 8.212252834472441096002849564389, 9.301178546757553526958267434221, 10.53622600687691340454258542756, 11.33359133996889118176969030818, 12.73146010216774514797104134533
