| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s − 4·11-s + 4·13-s + 15-s + 4·17-s − 8·19-s + 2·21-s + 8·23-s + 25-s + 27-s + 6·29-s − 31-s − 4·33-s + 2·35-s + 8·37-s + 4·39-s − 2·41-s − 4·43-s + 45-s + 2·47-s − 3·49-s + 4·51-s + 12·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 0.258·15-s + 0.970·17-s − 1.83·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.179·31-s − 0.696·33-s + 0.338·35-s + 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.609·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s + 0.560·51-s + 1.64·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.609555816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.609555816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.032123600855822364285851997846, −8.418474881322928277134227772178, −7.916436749199704423045179999737, −6.91925809736677649767298902581, −6.01895578997612513230565980572, −5.13165545919160353882204114778, −4.34428966495481972481180118758, −3.16585115055076990933891315817, −2.31100555017526674644461926506, −1.16205955578281514989555266229,
1.16205955578281514989555266229, 2.31100555017526674644461926506, 3.16585115055076990933891315817, 4.34428966495481972481180118758, 5.13165545919160353882204114778, 6.01895578997612513230565980572, 6.91925809736677649767298902581, 7.916436749199704423045179999737, 8.418474881322928277134227772178, 9.032123600855822364285851997846
