| L(s) = 1 | + 1.73·2-s + 0.999·4-s + 5-s − 1.73·8-s − 3·9-s + 1.73·10-s − 5.46·11-s + 2·13-s − 5·16-s + 3.46·17-s − 5.19·18-s − 2·19-s + 0.999·20-s − 9.46·22-s + 6.92·23-s + 25-s + 3.46·26-s + 29-s + 8.92·31-s − 5.19·32-s + 5.99·34-s − 2.99·36-s − 3.46·38-s − 1.73·40-s − 5.46·41-s + 3.46·43-s − 5.46·44-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s + 0.447·5-s − 0.612·8-s − 9-s + 0.547·10-s − 1.64·11-s + 0.554·13-s − 1.25·16-s + 0.840·17-s − 1.22·18-s − 0.458·19-s + 0.223·20-s − 2.01·22-s + 1.44·23-s + 0.200·25-s + 0.679·26-s + 0.185·29-s + 1.60·31-s − 0.918·32-s + 1.02·34-s − 0.499·36-s − 0.561·38-s − 0.273·40-s − 0.853·41-s + 0.528·43-s − 0.823·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.049131899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.049131899\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 4.92T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 1.46T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 1.46T + 79T^{2} \) |
| 83 | \( 1 - 1.07T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.913399817769832237921042076028, −7.05712100957669097187161233363, −6.12780661157302375599343746096, −5.75601590479250731972507535315, −5.07515500916270937368533768754, −4.60781403894123865048649320990, −3.42482563955295007436368372236, −2.91875126090396573346438931567, −2.29704650583877633179173676940, −0.70810508085074477967493143344,
0.70810508085074477967493143344, 2.29704650583877633179173676940, 2.91875126090396573346438931567, 3.42482563955295007436368372236, 4.60781403894123865048649320990, 5.07515500916270937368533768754, 5.75601590479250731972507535315, 6.12780661157302375599343746096, 7.05712100957669097187161233363, 7.913399817769832237921042076028
