| L(s) = 1 | − 2·5-s + 7-s + 2·11-s − 13-s + 6·17-s + 4·19-s + 2·23-s − 25-s − 4·31-s − 2·35-s + 2·37-s − 2·41-s − 4·43-s + 49-s + 12·53-s − 4·55-s + 6·61-s + 2·65-s − 12·67-s + 2·71-s − 2·73-s + 2·77-s + 4·83-s − 12·85-s − 2·89-s − 91-s − 8·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.603·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 0.417·23-s − 1/5·25-s − 0.718·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 1/7·49-s + 1.64·53-s − 0.539·55-s + 0.768·61-s + 0.248·65-s − 1.46·67-s + 0.237·71-s − 0.234·73-s + 0.227·77-s + 0.439·83-s − 1.30·85-s − 0.211·89-s − 0.104·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.846109562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.846109562\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.87972328959577902547586924639, −7.42232806638514620547046733388, −6.79842733527957873757569199492, −5.72146695164700154998523417106, −5.22205350947000092335589302969, −4.28559238960345030508806089721, −3.62987732862014125330273712536, −2.94719305713936828169314848733, −1.68443893879881686171875646777, −0.73319160908086279268180874527,
0.73319160908086279268180874527, 1.68443893879881686171875646777, 2.94719305713936828169314848733, 3.62987732862014125330273712536, 4.28559238960345030508806089721, 5.22205350947000092335589302969, 5.72146695164700154998523417106, 6.79842733527957873757569199492, 7.42232806638514620547046733388, 7.87972328959577902547586924639
