| L(s) = 1 | + 2.56·3-s + 5.20·7-s + 3.56·9-s + 0.899·11-s − 13-s + 5.66·17-s − 4.30·19-s + 13.3·21-s − 1.54·23-s + 1.43·27-s + 3.81·29-s − 6.12·31-s + 2.30·33-s + 3.12·37-s − 2.56·39-s − 11.0·41-s + 11.7·43-s + 9.00·47-s + 20.0·49-s + 14.5·51-s + 8.22·53-s − 11.0·57-s − 1.71·59-s − 10.2·61-s + 18.5·63-s − 6.43·67-s − 3.95·69-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 1.96·7-s + 1.18·9-s + 0.271·11-s − 0.277·13-s + 1.37·17-s − 0.987·19-s + 2.90·21-s − 0.321·23-s + 0.276·27-s + 0.708·29-s − 1.10·31-s + 0.401·33-s + 0.513·37-s − 0.410·39-s − 1.72·41-s + 1.79·43-s + 1.31·47-s + 2.87·49-s + 2.03·51-s + 1.13·53-s − 1.46·57-s − 0.223·59-s − 1.30·61-s + 2.33·63-s − 0.786·67-s − 0.476·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.618611305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.618611305\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 - 5.20T + 7T^{2} \) |
| 11 | \( 1 - 0.899T + 11T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 - 3.81T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 9.00T + 47T^{2} \) |
| 53 | \( 1 - 8.22T + 53T^{2} \) |
| 59 | \( 1 + 1.71T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 6.43T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 + 3.18T + 73T^{2} \) |
| 79 | \( 1 + 1.54T + 79T^{2} \) |
| 83 | \( 1 + 0.699T + 83T^{2} \) |
| 89 | \( 1 - 2.09T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−8.205075123610206993719439099240, −7.65994901771970624486313635949, −7.24657398580826595022790261567, −5.95150237818430410083056088193, −5.16841847027995920886199868222, −4.33174287118019407827792322878, −3.76933911319648399676630973704, −2.68948303874937942654962541324, −1.98889392972376236486549416518, −1.22240233496642961919264160443,
1.22240233496642961919264160443, 1.98889392972376236486549416518, 2.68948303874937942654962541324, 3.76933911319648399676630973704, 4.33174287118019407827792322878, 5.16841847027995920886199868222, 5.95150237818430410083056088193, 7.24657398580826595022790261567, 7.65994901771970624486313635949, 8.205075123610206993719439099240
