| L(s) = 1 | + 3-s − 5-s + 9-s − 2·11-s − 13-s − 15-s + 4·17-s + 19-s + 4·23-s + 25-s + 27-s + 5·31-s − 2·33-s − 5·37-s − 39-s − 2·41-s − 9·43-s − 45-s + 2·47-s + 4·51-s + 12·53-s + 2·55-s + 57-s + 8·59-s + 14·61-s + 65-s + 9·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.258·15-s + 0.970·17-s + 0.229·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.898·31-s − 0.348·33-s − 0.821·37-s − 0.160·39-s − 0.312·41-s − 1.37·43-s − 0.149·45-s + 0.291·47-s + 0.560·51-s + 1.64·53-s + 0.269·55-s + 0.132·57-s + 1.04·59-s + 1.79·61-s + 0.124·65-s + 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.057059357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.057059357\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.500459982889107791948072351983, −8.171950807751508916672808581419, −7.26411294117993636637552057855, −6.75853082244336358777403383052, −5.49024348463688223892917589201, −4.93300003544219171269645927792, −3.83714205863957343163612654118, −3.14294443805192686128138415785, −2.22601898104338938179843263310, −0.864305744297386540125109770640,
0.864305744297386540125109770640, 2.22601898104338938179843263310, 3.14294443805192686128138415785, 3.83714205863957343163612654118, 4.93300003544219171269645927792, 5.49024348463688223892917589201, 6.75853082244336358777403383052, 7.26411294117993636637552057855, 8.171950807751508916672808581419, 8.500459982889107791948072351983
