| L(s) = 1 | − 3-s − 4·7-s + 9-s − 3·11-s + 5·13-s + 7·17-s + 4·21-s − 9·23-s − 27-s − 7·31-s + 3·33-s − 8·37-s − 5·39-s + 3·41-s + 43-s − 8·47-s + 9·49-s − 7·51-s − 53-s − 8·59-s − 4·63-s + 9·67-s + 9·69-s + 12·71-s + 4·73-s + 12·77-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.904·11-s + 1.38·13-s + 1.69·17-s + 0.872·21-s − 1.87·23-s − 0.192·27-s − 1.25·31-s + 0.522·33-s − 1.31·37-s − 0.800·39-s + 0.468·41-s + 0.152·43-s − 1.16·47-s + 9/7·49-s − 0.980·51-s − 0.137·53-s − 1.04·59-s − 0.503·63-s + 1.09·67-s + 1.08·69-s + 1.42·71-s + 0.468·73-s + 1.36·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.27315003306405, −12.61422059752218, −12.39381022622650, −12.09776890443106, −11.32273364421886, −10.80436236109623, −10.54976711631553, −9.918288959607197, −9.635836464005135, −9.258232574593547, −8.307306926106777, −8.128821020943682, −7.578877695735375, −6.835610269914778, −6.557543528252461, −5.940790732150205, −5.509124959754802, −5.360681237415181, −4.312261457484706, −3.760933479647023, −3.408315383246707, −2.951412401805206, −2.042084016513315, −1.468881979583235, −0.6103063419062725, 0,
0.6103063419062725, 1.468881979583235, 2.042084016513315, 2.951412401805206, 3.408315383246707, 3.760933479647023, 4.312261457484706, 5.360681237415181, 5.509124959754802, 5.940790732150205, 6.557543528252461, 6.835610269914778, 7.578877695735375, 8.128821020943682, 8.307306926106777, 9.258232574593547, 9.635836464005135, 9.918288959607197, 10.54976711631553, 10.80436236109623, 11.32273364421886, 12.09776890443106, 12.39381022622650, 12.61422059752218, 13.27315003306405
