| L(s) = 1 | − 5-s − 4·7-s − 11-s + 6·17-s + 4·19-s + 4·23-s + 25-s − 2·29-s − 4·31-s + 4·35-s + 2·37-s − 8·41-s − 12·43-s + 9·49-s + 55-s − 4·59-s + 6·61-s − 6·67-s + 2·73-s + 4·77-s − 10·79-s − 6·83-s − 6·85-s − 6·89-s − 4·95-s − 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.301·11-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.676·35-s + 0.328·37-s − 1.24·41-s − 1.82·43-s + 9/7·49-s + 0.134·55-s − 0.520·59-s + 0.768·61-s − 0.733·67-s + 0.234·73-s + 0.455·77-s − 1.12·79-s − 0.658·83-s − 0.650·85-s − 0.635·89-s − 0.410·95-s − 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4652033501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4652033501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.56570411648731, −12.15512778597195, −11.77758409438328, −11.28546353408679, −10.74822664735392, −10.10128008749836, −9.956154558391402, −9.477295128273444, −9.012010452169191, −8.448373402003664, −7.941386487130716, −7.481803545605744, −6.904302490523606, −6.765459009719552, −5.995735171465309, −5.488711266274996, −5.215486077242923, −4.522922818799652, −3.736519577416371, −3.440426684039935, −3.042617061298385, −2.592258382501610, −1.566727846157463, −1.107122943628487, −0.1927136658665105,
0.1927136658665105, 1.107122943628487, 1.566727846157463, 2.592258382501610, 3.042617061298385, 3.440426684039935, 3.736519577416371, 4.522922818799652, 5.215486077242923, 5.488711266274996, 5.995735171465309, 6.765459009719552, 6.904302490523606, 7.481803545605744, 7.941386487130716, 8.448373402003664, 9.012010452169191, 9.477295128273444, 9.956154558391402, 10.10128008749836, 10.74822664735392, 11.28546353408679, 11.77758409438328, 12.15512778597195, 12.56570411648731
