| L(s) = 1 | − 5-s + 2·7-s − 3·11-s − 2·17-s − 19-s + 2·23-s + 25-s + 3·29-s + 3·31-s − 2·35-s − 5·41-s + 4·43-s − 8·47-s − 3·49-s − 2·53-s + 3·55-s + 3·59-s + 6·61-s + 10·67-s − 15·71-s − 14·73-s − 6·77-s + 8·79-s + 2·85-s − 89-s + 95-s − 16·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 0.904·11-s − 0.485·17-s − 0.229·19-s + 0.417·23-s + 1/5·25-s + 0.557·29-s + 0.538·31-s − 0.338·35-s − 0.780·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.404·55-s + 0.390·59-s + 0.768·61-s + 1.22·67-s − 1.78·71-s − 1.63·73-s − 0.683·77-s + 0.900·79-s + 0.216·85-s − 0.105·89-s + 0.102·95-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.78918236518773286913475765953, −7.01535680016112230896906091404, −6.33689815740263520027200536508, −5.34517461333505244110393027699, −4.81789313794149023057155377944, −4.13310115058146283469881208263, −3.12331532098106051333556860588, −2.35459814786668241019059005679, −1.29976458873617862099266512527, 0,
1.29976458873617862099266512527, 2.35459814786668241019059005679, 3.12331532098106051333556860588, 4.13310115058146283469881208263, 4.81789313794149023057155377944, 5.34517461333505244110393027699, 6.33689815740263520027200536508, 7.01535680016112230896906091404, 7.78918236518773286913475765953
