| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 11-s + 2·13-s + 16-s + 6·17-s + 5·19-s + 20-s + 22-s − 3·23-s − 4·25-s + 2·26-s − 2·29-s − 5·31-s + 32-s + 6·34-s + 3·37-s + 5·38-s + 40-s − 3·41-s − 2·43-s + 44-s − 3·46-s + 10·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s + 1/4·16-s + 1.45·17-s + 1.14·19-s + 0.223·20-s + 0.213·22-s − 0.625·23-s − 4/5·25-s + 0.392·26-s − 0.371·29-s − 0.898·31-s + 0.176·32-s + 1.02·34-s + 0.493·37-s + 0.811·38-s + 0.158·40-s − 0.468·41-s − 0.304·43-s + 0.150·44-s − 0.442·46-s + 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.488509707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.488509707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.902652306745800629688185317935, −7.87708987641543103898693651601, −7.35884172845471818919501955979, −6.32945863492244277756978418003, −5.68059987312629711333716968262, −5.14077577262328240886672746588, −3.89184467651147871296247551688, −3.38941718058539752408277411208, −2.19137305889152531239902913655, −1.15163143868737354155920287613,
1.15163143868737354155920287613, 2.19137305889152531239902913655, 3.38941718058539752408277411208, 3.89184467651147871296247551688, 5.14077577262328240886672746588, 5.68059987312629711333716968262, 6.32945863492244277756978418003, 7.35884172845471818919501955979, 7.87708987641543103898693651601, 8.902652306745800629688185317935
