| L(s) = 1 | − 2·4-s + 5-s + 1.73·7-s + 4·16-s + 3.46·17-s − 5.19·19-s − 2·20-s − 6·23-s + 25-s − 3.46·28-s − 6.92·29-s + 31-s + 1.73·35-s − 5·37-s − 3.46·41-s + 10.3·43-s + 12·47-s − 4·49-s − 6·53-s − 12.1·61-s − 8·64-s − 5·67-s − 6.92·68-s + 6·71-s − 1.73·73-s + 10.3·76-s − 15.5·79-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.654·7-s + 16-s + 0.840·17-s − 1.19·19-s − 0.447·20-s − 1.25·23-s + 0.200·25-s − 0.654·28-s − 1.28·29-s + 0.179·31-s + 0.292·35-s − 0.821·37-s − 0.541·41-s + 1.58·43-s + 1.75·47-s − 0.571·49-s − 0.824·53-s − 1.55·61-s − 64-s − 0.610·67-s − 0.840·68-s + 0.712·71-s − 0.202·73-s + 1.19·76-s − 1.75·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 1.73T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 13T + 97T^{2} \) |
| 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
−7.894861231744188535033955422134, −7.30118650248473850185636859406, −6.06095472210945953963578523424, −5.71758177053910900655005878607, −4.82379238813294573062442134303, −4.19810554261176928542830850020, −3.44488025490452498976496974510, −2.22600041737460393263473731783, −1.34914035465900881286716922310, 0,
1.34914035465900881286716922310, 2.22600041737460393263473731783, 3.44488025490452498976496974510, 4.19810554261176928542830850020, 4.82379238813294573062442134303, 5.71758177053910900655005878607, 6.06095472210945953963578523424, 7.30118650248473850185636859406, 7.894861231744188535033955422134
