| L(s) = 1 | − 5·5-s − 14.5·7-s − 49.2·11-s + 72.1·13-s − 118.·17-s − 123.·19-s − 91.4·23-s + 25·25-s − 174.·29-s + 46.2·31-s + 72.5·35-s + 154.·37-s + 364.·41-s − 125.·43-s − 221.·47-s − 132.·49-s + 13.6·53-s + 246.·55-s + 239.·59-s − 54.5·61-s − 360.·65-s + 76.0·67-s + 728.·71-s − 501.·73-s + 715.·77-s − 397.·79-s − 1.36e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.783·7-s − 1.35·11-s + 1.53·13-s − 1.68·17-s − 1.48·19-s − 0.829·23-s + 0.200·25-s − 1.11·29-s + 0.268·31-s + 0.350·35-s + 0.688·37-s + 1.38·41-s − 0.445·43-s − 0.687·47-s − 0.385·49-s + 0.0354·53-s + 0.604·55-s + 0.527·59-s − 0.114·61-s − 0.688·65-s + 0.138·67-s + 1.21·71-s − 0.804·73-s + 1.05·77-s − 0.566·79-s − 1.81·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6218994256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6218994256\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5T \) |
| good | 7 | \( 1 + 14.5T + 343T^{2} \) |
| 11 | \( 1 + 49.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 91.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 46.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 364.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 221.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 13.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 239.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 76.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 728.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 501.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 397.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.36e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 335.T + 9.12e5T^{2} \) |
| 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
−8.525200820273198151800212692527, −8.174719679086535478926035390060, −7.10716059641723724604890786719, −6.32070761248735785846845142379, −5.76217083934571988092499013672, −4.48742436322822095429510331479, −3.90429191007281031337615339753, −2.84883472603925296204039487719, −1.93904557679539961835098877897, −0.33686646246872973884210936249,
0.33686646246872973884210936249, 1.93904557679539961835098877897, 2.84883472603925296204039487719, 3.90429191007281031337615339753, 4.48742436322822095429510331479, 5.76217083934571988092499013672, 6.32070761248735785846845142379, 7.10716059641723724604890786719, 8.174719679086535478926035390060, 8.525200820273198151800212692527
