| L(s) = 1 | − 3·9-s + 11-s + 2·13-s + 4·17-s + 2·19-s − 5·23-s + 29-s + 2·31-s − 3·37-s − 12·41-s − 11·43-s + 2·47-s − 6·53-s + 10·59-s − 4·61-s − 67-s − 3·71-s − 9·79-s + 9·81-s − 2·83-s + 6·89-s + 14·97-s − 3·99-s − 12·101-s + 14·103-s + 12·107-s + 5·109-s + ⋯ |
| L(s) = 1 | − 9-s + 0.301·11-s + 0.554·13-s + 0.970·17-s + 0.458·19-s − 1.04·23-s + 0.185·29-s + 0.359·31-s − 0.493·37-s − 1.87·41-s − 1.67·43-s + 0.291·47-s − 0.824·53-s + 1.30·59-s − 0.512·61-s − 0.122·67-s − 0.356·71-s − 1.01·79-s + 81-s − 0.219·83-s + 0.635·89-s + 1.42·97-s − 0.301·99-s − 1.19·101-s + 1.37·103-s + 1.16·107-s + 0.478·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.35157268044000145956615981069, −6.55993020930790143884579944137, −5.96641061285835348653219528541, −5.35042178391152919502569971744, −4.64728964037147877560900706703, −3.51146178172572412976933678924, −3.27888947146103744610140058793, −2.13680609870815444333070671956, −1.23394157265824250732773943069, 0,
1.23394157265824250732773943069, 2.13680609870815444333070671956, 3.27888947146103744610140058793, 3.51146178172572412976933678924, 4.64728964037147877560900706703, 5.35042178391152919502569971744, 5.96641061285835348653219528541, 6.55993020930790143884579944137, 7.35157268044000145956615981069
