| L(s) = 1 | + 5-s + 6·13-s + 7·17-s + 4·19-s − 3·23-s + 25-s + 4·29-s + 3·31-s − 2·41-s − 4·43-s + 7·47-s − 7·49-s + 5·53-s − 4·59-s − 61-s + 6·65-s + 2·67-s + 10·71-s + 4·73-s + 5·79-s + 12·83-s + 7·85-s + 4·95-s + 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.66·13-s + 1.69·17-s + 0.917·19-s − 0.625·23-s + 1/5·25-s + 0.742·29-s + 0.538·31-s − 0.312·41-s − 0.609·43-s + 1.02·47-s − 49-s + 0.686·53-s − 0.520·59-s − 0.128·61-s + 0.744·65-s + 0.244·67-s + 1.18·71-s + 0.468·73-s + 0.562·79-s + 1.31·83-s + 0.759·85-s + 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.437045437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.437045437\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.58586066264879, −15.06093845297974, −14.27603913060401, −13.87290726905469, −13.59068307696153, −12.86090747772532, −12.19799518471220, −11.82364314316323, −11.15762969675313, −10.49655105467013, −10.07875802641084, −9.488103799359001, −8.909272471023496, −8.044272190381600, −7.985826901384936, −6.964730676212692, −6.365753145909560, −5.798223908641816, −5.320581637632740, −4.546436213044026, −3.533747021958977, −3.359875095190327, −2.324243452290875, −1.358654351162996, −0.8625655939945225,
0.8625655939945225, 1.358654351162996, 2.324243452290875, 3.359875095190327, 3.533747021958977, 4.546436213044026, 5.320581637632740, 5.798223908641816, 6.365753145909560, 6.964730676212692, 7.985826901384936, 8.044272190381600, 8.909272471023496, 9.488103799359001, 10.07875802641084, 10.49655105467013, 11.15762969675313, 11.82364314316323, 12.19799518471220, 12.86090747772532, 13.59068307696153, 13.87290726905469, 14.27603913060401, 15.06093845297974, 15.58586066264879
