L(s) = 1 | + 5-s + 2·7-s − 2·11-s + 2·13-s − 8·17-s + 2·19-s − 6·23-s + 25-s + 6·29-s + 2·35-s + 10·37-s − 2·41-s − 43-s − 6·47-s − 3·49-s − 2·53-s − 2·55-s + 10·59-s + 4·61-s + 2·65-s − 4·67-s − 8·73-s − 4·77-s − 8·79-s + 4·83-s − 8·85-s − 6·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.603·11-s + 0.554·13-s − 1.94·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s + 0.338·35-s + 1.64·37-s − 0.312·41-s − 0.152·43-s − 0.875·47-s − 3/7·49-s − 0.274·53-s − 0.269·55-s + 1.30·59-s + 0.512·61-s + 0.248·65-s − 0.488·67-s − 0.936·73-s − 0.455·77-s − 0.900·79-s + 0.439·83-s − 0.867·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.45011910927979, −14.67152237659311, −14.41498198016470, −13.64036168023243, −13.31291276121592, −12.93947986792560, −12.09647831196629, −11.50560136878898, −11.18407496439464, −10.56687946506426, −10.01367575586574, −9.488150201675707, −8.748101077306825, −8.286961856866810, −7.900463751859182, −7.040809968328881, −6.494658664936517, −5.945048907764376, −5.303337867026506, −4.528709951002333, −4.293954650455140, −3.251001197965821, −2.465177893713714, −1.957387750027457, −1.113794848308441, 0,
1.113794848308441, 1.957387750027457, 2.465177893713714, 3.251001197965821, 4.293954650455140, 4.528709951002333, 5.303337867026506, 5.945048907764376, 6.494658664936517, 7.040809968328881, 7.900463751859182, 8.286961856866810, 8.748101077306825, 9.488150201675707, 10.01367575586574, 10.56687946506426, 11.18407496439464, 11.50560136878898, 12.09647831196629, 12.93947986792560, 13.31291276121592, 13.64036168023243, 14.41498198016470, 14.67152237659311, 15.45011910927979