| L(s) = 1 | − 3-s − 2·9-s + 11-s + 2·13-s + 3·17-s − 3·19-s − 6·23-s + 5·27-s + 2·29-s − 2·31-s − 33-s − 4·37-s − 2·39-s − 3·41-s + 4·43-s + 6·47-s − 7·49-s − 3·51-s + 10·53-s + 3·57-s − 12·59-s + 12·61-s + 67-s + 6·69-s − 10·71-s + 73-s − 16·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.301·11-s + 0.554·13-s + 0.727·17-s − 0.688·19-s − 1.25·23-s + 0.962·27-s + 0.371·29-s − 0.359·31-s − 0.174·33-s − 0.657·37-s − 0.320·39-s − 0.468·41-s + 0.609·43-s + 0.875·47-s − 49-s − 0.420·51-s + 1.37·53-s + 0.397·57-s − 1.56·59-s + 1.53·61-s + 0.122·67-s + 0.722·69-s − 1.18·71-s + 0.117·73-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.437902216116457195185806352436, −7.53457795797380300974441515672, −6.64548430672796580839894192397, −5.93810049826618219908085515030, −5.45207291476929754602689232535, −4.39164678241329634586175014939, −3.60233517849484739120590881039, −2.56940418914897998415786357379, −1.36426937886750358192679635235, 0,
1.36426937886750358192679635235, 2.56940418914897998415786357379, 3.60233517849484739120590881039, 4.39164678241329634586175014939, 5.45207291476929754602689232535, 5.93810049826618219908085515030, 6.64548430672796580839894192397, 7.53457795797380300974441515672, 8.437902216116457195185806352436
