| L(s) = 1 | − 1.23·3-s + 3.23·5-s − 7-s − 1.47·9-s + 6.47·11-s + 0.763·13-s − 4.00·15-s + 4.47·17-s + 1.23·19-s + 1.23·21-s − 4·23-s + 5.47·25-s + 5.52·27-s − 4.47·29-s + 2.47·31-s − 8.00·33-s − 3.23·35-s − 4.47·37-s − 0.944·39-s − 8.47·41-s − 6.47·43-s − 4.76·45-s − 10.4·47-s + 49-s − 5.52·51-s − 10·53-s + 20.9·55-s + ⋯ |
| L(s) = 1 | − 0.713·3-s + 1.44·5-s − 0.377·7-s − 0.490·9-s + 1.95·11-s + 0.211·13-s − 1.03·15-s + 1.08·17-s + 0.283·19-s + 0.269·21-s − 0.834·23-s + 1.09·25-s + 1.06·27-s − 0.830·29-s + 0.444·31-s − 1.39·33-s − 0.546·35-s − 0.735·37-s − 0.151·39-s − 1.32·41-s − 0.986·43-s − 0.710·45-s − 1.52·47-s + 0.142·49-s − 0.774·51-s − 1.37·53-s + 2.82·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.228131228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228131228\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 6.47T + 11T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−12.05972195931707119663777641652, −11.47458123413300220597019137645, −10.12554122039553452308402962481, −9.572373527065341565451768546361, −8.524978190737606320675831669004, −6.71752546290169180092862320997, −6.11868239118300673563659650682, −5.23651453418504227026658067374, −3.48053718942393294634255816722, −1.57060045378259883863154228778,
1.57060045378259883863154228778, 3.48053718942393294634255816722, 5.23651453418504227026658067374, 6.11868239118300673563659650682, 6.71752546290169180092862320997, 8.524978190737606320675831669004, 9.572373527065341565451768546361, 10.12554122039553452308402962481, 11.47458123413300220597019137645, 12.05972195931707119663777641652
