L(s) = 1 | − 3-s − 5-s + 7-s − 2·9-s − 5·13-s + 15-s − 3·17-s + 2·19-s − 21-s + 6·23-s + 25-s + 5·27-s − 3·29-s + 4·31-s − 35-s + 2·37-s + 5·39-s + 12·41-s − 10·43-s + 2·45-s − 9·47-s + 49-s + 3·51-s + 12·53-s − 2·57-s − 8·61-s − 2·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.38·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s − 0.218·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s + 0.800·39-s + 1.87·41-s − 1.52·43-s + 0.298·45-s − 1.31·47-s + 1/7·49-s + 0.420·51-s + 1.64·53-s − 0.264·57-s − 1.02·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9537107077\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9537107077\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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
−14.24693011878691, −13.69648856622150, −13.06775347659684, −12.63555847478761, −12.02329012145661, −11.62038494371479, −11.27757819489321, −10.78376062730788, −10.21377841462648, −9.569241963943306, −9.098328284967463, −8.530488289028281, −7.951375453916034, −7.448388462856904, −6.888293816171589, −6.431868907244784, −5.647209694957754, −5.166103078219440, −4.719899708184433, −4.210569065855972, −3.276943354165636, −2.749556234913048, −2.153525597603477, −1.135932407218472, −0.3726268198924754,
0.3726268198924754, 1.135932407218472, 2.153525597603477, 2.749556234913048, 3.276943354165636, 4.210569065855972, 4.719899708184433, 5.166103078219440, 5.647209694957754, 6.431868907244784, 6.888293816171589, 7.448388462856904, 7.951375453916034, 8.530488289028281, 9.098328284967463, 9.569241963943306, 10.21377841462648, 10.78376062730788, 11.27757819489321, 11.62038494371479, 12.02329012145661, 12.63555847478761, 13.06775347659684, 13.69648856622150, 14.24693011878691