| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 3·11-s − 6·13-s + 2·14-s + 16-s + 2·17-s − 19-s − 3·22-s + 23-s + 6·26-s − 2·28-s + 5·29-s + 7·31-s − 32-s − 2·34-s − 2·37-s + 38-s − 2·41-s − 6·43-s + 3·44-s − 46-s + 12·47-s − 3·49-s − 6·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.904·11-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.639·22-s + 0.208·23-s + 1.17·26-s − 0.377·28-s + 0.928·29-s + 1.25·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s + 0.162·38-s − 0.312·41-s − 0.914·43-s + 0.452·44-s − 0.147·46-s + 1.75·47-s − 3/7·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.022008289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.022008289\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.78988027885184169255078047607, −7.05938521245110062062396355988, −6.61111728234922558511790007770, −5.93398290478062145853210018626, −4.97731232615098753538443603999, −4.28388318353640632027476003581, −3.21614458958718745996877270534, −2.67008121620986198516879531453, −1.63644234327442444844396675140, −0.54960597947417733630301958269,
0.54960597947417733630301958269, 1.63644234327442444844396675140, 2.67008121620986198516879531453, 3.21614458958718745996877270534, 4.28388318353640632027476003581, 4.97731232615098753538443603999, 5.93398290478062145853210018626, 6.61111728234922558511790007770, 7.05938521245110062062396355988, 7.78988027885184169255078047607
