| L(s) = 1 | − 2·4-s + 5-s + 7-s + 3·11-s + 5·13-s + 4·16-s − 3·17-s + 2·19-s − 2·20-s + 6·23-s + 25-s − 2·28-s − 3·29-s − 4·31-s + 35-s + 2·37-s + 12·41-s − 10·43-s − 6·44-s − 9·47-s + 49-s − 10·52-s − 12·53-s + 3·55-s + 8·61-s − 8·64-s + 5·65-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.377·7-s + 0.904·11-s + 1.38·13-s + 16-s − 0.727·17-s + 0.458·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.377·28-s − 0.557·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s + 1.87·41-s − 1.52·43-s − 0.904·44-s − 1.31·47-s + 1/7·49-s − 1.38·52-s − 1.64·53-s + 0.404·55-s + 1.02·61-s − 64-s + 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.273082906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273082906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−11.50769791186352205274303435148, −10.81642134200306905635056067311, −9.502060864672765915422726581101, −9.001872818363131182920485953111, −8.087023964728641191414598204496, −6.69771158952637091109528432278, −5.63914839037632160871019437372, −4.54321060660836499326534703030, −3.45067056060882590557565434961, −1.35738749247200780270220785705,
1.35738749247200780270220785705, 3.45067056060882590557565434961, 4.54321060660836499326534703030, 5.63914839037632160871019437372, 6.69771158952637091109528432278, 8.087023964728641191414598204496, 9.001872818363131182920485953111, 9.502060864672765915422726581101, 10.81642134200306905635056067311, 11.50769791186352205274303435148
