| L(s) = 1 | − 5-s + 2·13-s − 6·17-s + 4·19-s + 8·23-s + 25-s + 2·29-s − 4·31-s + 10·37-s − 2·41-s + 4·43-s + 8·47-s − 7·49-s + 2·53-s + 8·59-s − 2·61-s − 2·65-s + 12·67-s + 8·71-s − 14·73-s + 12·79-s − 4·83-s + 6·85-s + 14·89-s − 4·95-s + 2·97-s + 10·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 1.64·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s − 49-s + 0.274·53-s + 1.04·59-s − 0.256·61-s − 0.248·65-s + 1.46·67-s + 0.949·71-s − 1.63·73-s + 1.35·79-s − 0.439·83-s + 0.650·85-s + 1.48·89-s − 0.410·95-s + 0.203·97-s + 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.549221111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.549221111\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.307341106649090313623323741633, −8.855778251487975110890093783326, −7.903252653224454437423095775636, −7.10466480215322585099594808810, −6.37297543125638649107463074905, −5.29529188897154455060743121312, −4.44715554925080312846390934231, −3.49286559891962741437411069128, −2.44107391193880515870491532764, −0.914946353496096151316831566481,
0.914946353496096151316831566481, 2.44107391193880515870491532764, 3.49286559891962741437411069128, 4.44715554925080312846390934231, 5.29529188897154455060743121312, 6.37297543125638649107463074905, 7.10466480215322585099594808810, 7.903252653224454437423095775636, 8.855778251487975110890093783326, 9.307341106649090313623323741633
