| L(s) = 1 | − 3-s − 5-s + 7-s − 2·9-s − 4·11-s − 4·13-s + 15-s + 4·17-s + 19-s − 21-s − 4·23-s − 4·25-s + 5·27-s − 3·29-s − 2·31-s + 4·33-s − 35-s + 8·37-s + 4·39-s − 2·41-s + 8·43-s + 2·45-s − 47-s + 49-s − 4·51-s + 8·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.20·11-s − 1.10·13-s + 0.258·15-s + 0.970·17-s + 0.229·19-s − 0.218·21-s − 0.834·23-s − 4/5·25-s + 0.962·27-s − 0.557·29-s − 0.359·31-s + 0.696·33-s − 0.169·35-s + 1.31·37-s + 0.640·39-s − 0.312·41-s + 1.21·43-s + 0.298·45-s − 0.145·47-s + 1/7·49-s − 0.560·51-s + 1.09·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.19045401658049, −13.44764790345246, −12.99833534336357, −12.32443616392812, −11.95223524737756, −11.76927430563300, −10.95030535297755, −10.72239759371306, −10.15415893781042, −9.567113349674381, −9.173874205706664, −8.303125477815227, −7.864776027022198, −7.636785957385236, −7.119255776495497, −6.219222075266591, −5.643024465334305, −5.518185181625404, −4.749799634232581, −4.333222779065209, −3.548068254138568, −2.848595700892036, −2.423286223159660, −1.616009115708318, −0.6049027137249306, 0,
0.6049027137249306, 1.616009115708318, 2.423286223159660, 2.848595700892036, 3.548068254138568, 4.333222779065209, 4.749799634232581, 5.518185181625404, 5.643024465334305, 6.219222075266591, 7.119255776495497, 7.636785957385236, 7.864776027022198, 8.303125477815227, 9.173874205706664, 9.567113349674381, 10.15415893781042, 10.72239759371306, 10.95030535297755, 11.76927430563300, 11.95223524737756, 12.32443616392812, 12.99833534336357, 13.44764790345246, 14.19045401658049
