| L(s) = 1 | + 3.73·5-s + 3.46·7-s + 2·11-s − 2.46·13-s + 2.26·17-s − 7.46·19-s − 4.92·23-s + 8.92·25-s + 4.26·29-s − 10.9·31-s + 12.9·35-s − 0.464·37-s + 6.92·41-s − 4.53·43-s + 6.92·47-s + 4.99·49-s − 10.9·53-s + 7.46·55-s + 8·59-s + 10.4·61-s − 9.19·65-s − 0.535·67-s − 2·71-s + 73-s + 6.92·77-s + 0.535·79-s − 2.92·83-s + ⋯ |
| L(s) = 1 | + 1.66·5-s + 1.30·7-s + 0.603·11-s − 0.683·13-s + 0.550·17-s − 1.71·19-s − 1.02·23-s + 1.78·25-s + 0.792·29-s − 1.96·31-s + 2.18·35-s − 0.0762·37-s + 1.08·41-s − 0.691·43-s + 1.01·47-s + 0.714·49-s − 1.50·53-s + 1.00·55-s + 1.04·59-s + 1.33·61-s − 1.14·65-s − 0.0654·67-s − 0.237·71-s + 0.117·73-s + 0.789·77-s + 0.0602·79-s − 0.321·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.178557401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.178557401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 0.464T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 0.535T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 0.535T + 79T^{2} \) |
| 83 | \( 1 + 2.92T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
| 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
−10.48584878051980350058024399621, −9.732447933257023829202295277555, −8.900697090684450437226288834605, −8.053902988172979841085948467782, −6.89000206883932908924473476369, −5.94514531753934171188745194498, −5.19539501856225733376738619508, −4.16345793561819089967781572084, −2.33773378312612493231361164219, −1.61641617724557270702584577553,
1.61641617724557270702584577553, 2.33773378312612493231361164219, 4.16345793561819089967781572084, 5.19539501856225733376738619508, 5.94514531753934171188745194498, 6.89000206883932908924473476369, 8.053902988172979841085948467782, 8.900697090684450437226288834605, 9.732447933257023829202295277555, 10.48584878051980350058024399621
