| L(s) = 1 | − 5-s − 4·11-s − 2·13-s − 2·17-s − 4·19-s + 25-s + 2·29-s − 10·37-s − 10·41-s − 4·43-s + 8·47-s − 7·49-s + 10·53-s + 4·55-s − 4·59-s − 2·61-s + 2·65-s − 12·67-s − 8·71-s + 10·73-s + 12·83-s + 2·85-s + 6·89-s + 4·95-s + 2·97-s − 6·101-s + 16·103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s − 1.64·37-s − 1.56·41-s − 0.609·43-s + 1.16·47-s − 49-s + 1.37·53-s + 0.539·55-s − 0.520·59-s − 0.256·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s + 1.17·73-s + 1.31·83-s + 0.216·85-s + 0.635·89-s + 0.410·95-s + 0.203·97-s − 0.597·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.26700477708582759405639390127, −8.982199030019329801304860528684, −8.270140448885357284476919824336, −7.40264879408133961073539513214, −6.55157891837643531715592264808, −5.32469243442401148699352691061, −4.54153907278640853760279816056, −3.29035819756263414840801407793, −2.11935007719153584295308930473, 0,
2.11935007719153584295308930473, 3.29035819756263414840801407793, 4.54153907278640853760279816056, 5.32469243442401148699352691061, 6.55157891837643531715592264808, 7.40264879408133961073539513214, 8.270140448885357284476919824336, 8.982199030019329801304860528684, 10.26700477708582759405639390127
