| L(s) = 1 | + 5-s − 3·9-s + 11-s + 2·13-s + 6·17-s + 4·19-s − 4·23-s + 25-s + 6·29-s + 8·31-s − 2·37-s + 2·41-s − 4·43-s − 3·45-s + 12·47-s − 7·49-s − 2·53-s + 55-s − 4·59-s − 10·61-s + 2·65-s + 16·67-s − 8·71-s + 14·73-s − 8·79-s + 9·81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s + 0.301·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.75·47-s − 49-s − 0.274·53-s + 0.134·55-s − 0.520·59-s − 1.28·61-s + 0.248·65-s + 1.95·67-s − 0.949·71-s + 1.63·73-s − 0.900·79-s + 81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.721273448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.721273448\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.07588915028781859505118646541, −9.376036497146997809466572379928, −8.402071812243779349662792300606, −7.78041223670494609407513138154, −6.49908708730306794178412805091, −5.83555924118533390298503875284, −4.97617253141282559040022801022, −3.60799086718076388676488423337, −2.69303075235727048947048138558, −1.13555459814448309624758433433,
1.13555459814448309624758433433, 2.69303075235727048947048138558, 3.60799086718076388676488423337, 4.97617253141282559040022801022, 5.83555924118533390298503875284, 6.49908708730306794178412805091, 7.78041223670494609407513138154, 8.402071812243779349662792300606, 9.376036497146997809466572379928, 10.07588915028781859505118646541
