| L(s) = 1 | − 5·3-s − 10·7-s − 2·9-s − 15·11-s − 8·13-s − 21·17-s + 105·19-s + 50·21-s − 10·23-s + 145·27-s + 20·29-s + 230·31-s + 75·33-s + 54·37-s + 40·39-s − 195·41-s + 300·43-s − 480·47-s − 243·49-s + 105·51-s − 322·53-s − 525·57-s + 560·59-s + 730·61-s + 20·63-s − 255·67-s + 50·69-s + ⋯ |
| L(s) = 1 | − 0.962·3-s − 0.539·7-s − 0.0740·9-s − 0.411·11-s − 0.170·13-s − 0.299·17-s + 1.26·19-s + 0.519·21-s − 0.0906·23-s + 1.03·27-s + 0.128·29-s + 1.33·31-s + 0.395·33-s + 0.239·37-s + 0.164·39-s − 0.742·41-s + 1.06·43-s − 1.48·47-s − 0.708·49-s + 0.288·51-s − 0.834·53-s − 1.21·57-s + 1.23·59-s + 1.53·61-s + 0.0399·63-s − 0.464·67-s + 0.0872·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 10 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 + 8 T + p^{3} T^{2} \) |
| 17 | \( 1 + 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 105 T + p^{3} T^{2} \) |
| 23 | \( 1 + 10 T + p^{3} T^{2} \) |
| 29 | \( 1 - 20 T + p^{3} T^{2} \) |
| 31 | \( 1 - 230 T + p^{3} T^{2} \) |
| 37 | \( 1 - 54 T + p^{3} T^{2} \) |
| 41 | \( 1 + 195 T + p^{3} T^{2} \) |
| 43 | \( 1 - 300 T + p^{3} T^{2} \) |
| 47 | \( 1 + 480 T + p^{3} T^{2} \) |
| 53 | \( 1 + 322 T + p^{3} T^{2} \) |
| 59 | \( 1 - 560 T + p^{3} T^{2} \) |
| 61 | \( 1 - 730 T + p^{3} T^{2} \) |
| 67 | \( 1 + 255 T + p^{3} T^{2} \) |
| 71 | \( 1 - 40 T + p^{3} T^{2} \) |
| 73 | \( 1 - 317 T + p^{3} T^{2} \) |
| 79 | \( 1 - 830 T + p^{3} T^{2} \) |
| 83 | \( 1 + 75 T + p^{3} T^{2} \) |
| 89 | \( 1 + 705 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1434 T + p^{3} T^{2} \) |
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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
−8.615650214058797700014521079337, −7.81634356780046803782332320707, −6.81710326710241891567019317139, −6.22248550819816178047226961974, −5.34741108350338251659500412109, −4.73253242633978215888736634938, −3.45460285330901364516786104721, −2.54338124487252833675581526659, −1.01483017898074344975299473130, 0,
1.01483017898074344975299473130, 2.54338124487252833675581526659, 3.45460285330901364516786104721, 4.73253242633978215888736634938, 5.34741108350338251659500412109, 6.22248550819816178047226961974, 6.81710326710241891567019317139, 7.81634356780046803782332320707, 8.615650214058797700014521079337
