| L(s) = 1 | − 3-s + 7-s − 2·9-s + 3·11-s + 13-s + 7·17-s − 21-s − 6·23-s + 5·27-s − 5·29-s − 2·31-s − 3·33-s + 2·37-s − 39-s + 2·41-s + 4·43-s + 3·47-s + 49-s − 7·51-s + 6·53-s − 10·59-s − 8·61-s − 2·63-s − 2·67-s + 6·69-s + 8·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.277·13-s + 1.69·17-s − 0.218·21-s − 1.25·23-s + 0.962·27-s − 0.928·29-s − 0.359·31-s − 0.522·33-s + 0.328·37-s − 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.437·47-s + 1/7·49-s − 0.980·51-s + 0.824·53-s − 1.30·59-s − 1.02·61-s − 0.251·63-s − 0.244·67-s + 0.722·69-s + 0.949·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.528951675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.528951675\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.846837941585832497895972630746, −7.943950985696702485922224990231, −7.41192519294123246216805168593, −6.20892833382707396477507471574, −5.86948996746859435559692437938, −5.04950450614587838523509099308, −4.01558432174714569962208636972, −3.25831654996759112584151679894, −1.95044579129989398832305438850, −0.808304732563204183922021296054,
0.808304732563204183922021296054, 1.95044579129989398832305438850, 3.25831654996759112584151679894, 4.01558432174714569962208636972, 5.04950450614587838523509099308, 5.86948996746859435559692437938, 6.20892833382707396477507471574, 7.41192519294123246216805168593, 7.943950985696702485922224990231, 8.846837941585832497895972630746
