| L(s) = 1 | − 2·3-s + 2·5-s + 9-s − 11-s − 2·13-s − 4·15-s − 2·19-s − 25-s + 4·27-s + 6·29-s + 4·31-s + 2·33-s + 2·37-s + 4·39-s − 8·41-s − 12·43-s + 2·45-s + 12·47-s − 2·53-s − 2·55-s + 4·57-s + 10·59-s + 10·61-s − 4·65-s + 12·67-s − 4·71-s − 12·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 1.03·15-s − 0.458·19-s − 1/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.348·33-s + 0.328·37-s + 0.640·39-s − 1.24·41-s − 1.82·43-s + 0.298·45-s + 1.75·47-s − 0.274·53-s − 0.269·55-s + 0.529·57-s + 1.30·59-s + 1.28·61-s − 0.496·65-s + 1.46·67-s − 0.474·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.12460300693083926763643198058, −6.62617679408616128843528252117, −6.01528735818747959290218806405, −5.36328973351482376314419255581, −4.92734604444592399564955127816, −4.07580840862266141599266673735, −2.89616711825603720890335339875, −2.17854992458334996126947913131, −1.11046195028346012436707118039, 0,
1.11046195028346012436707118039, 2.17854992458334996126947913131, 2.89616711825603720890335339875, 4.07580840862266141599266673735, 4.92734604444592399564955127816, 5.36328973351482376314419255581, 6.01528735818747959290218806405, 6.62617679408616128843528252117, 7.12460300693083926763643198058
