| L(s) = 1 | − 1.96·3-s − 2.74·5-s − 4.75·7-s + 0.854·9-s − 4.54·11-s + 2.89·13-s + 5.39·15-s + 17-s − 0.411·19-s + 9.34·21-s − 8.30·23-s + 2.55·25-s + 4.21·27-s + 9.19·29-s − 3.50·31-s + 8.93·33-s + 13.0·35-s − 8.19·37-s − 5.68·39-s + 0.0614·41-s − 5.65·43-s − 2.34·45-s − 5.64·47-s + 15.6·49-s − 1.96·51-s + 0.672·53-s + 12.5·55-s + ⋯ |
| L(s) = 1 | − 1.13·3-s − 1.22·5-s − 1.79·7-s + 0.284·9-s − 1.37·11-s + 0.803·13-s + 1.39·15-s + 0.242·17-s − 0.0943·19-s + 2.03·21-s − 1.73·23-s + 0.510·25-s + 0.810·27-s + 1.70·29-s − 0.628·31-s + 1.55·33-s + 2.21·35-s − 1.34·37-s − 0.910·39-s + 0.00960·41-s − 0.862·43-s − 0.350·45-s − 0.824·47-s + 2.23·49-s − 0.274·51-s + 0.0923·53-s + 1.68·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.96T + 3T^{2} \) |
| 5 | \( 1 + 2.74T + 5T^{2} \) |
| 7 | \( 1 + 4.75T + 7T^{2} \) |
| 11 | \( 1 + 4.54T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 19 | \( 1 + 0.411T + 19T^{2} \) |
| 23 | \( 1 + 8.30T + 23T^{2} \) |
| 29 | \( 1 - 9.19T + 29T^{2} \) |
| 31 | \( 1 + 3.50T + 31T^{2} \) |
| 37 | \( 1 + 8.19T + 37T^{2} \) |
| 41 | \( 1 - 0.0614T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + 5.64T + 47T^{2} \) |
| 53 | \( 1 - 0.672T + 53T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 7.24T + 89T^{2} \) |
| 97 | \( 1 - 9.23T + 97T^{2} \) |
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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
−7.35782145293732907871513906110, −6.61301301439246686141060657336, −6.21460874021617194721428374663, −5.46174340464512638246533193173, −4.79105396251892554765622537938, −3.68634859336512450516915720317, −3.42465846356370783979863022395, −2.36593916497898842042657257153, −0.63524114619909775004522441308, 0,
0.63524114619909775004522441308, 2.36593916497898842042657257153, 3.42465846356370783979863022395, 3.68634859336512450516915720317, 4.79105396251892554765622537938, 5.46174340464512638246533193173, 6.21460874021617194721428374663, 6.61301301439246686141060657336, 7.35782145293732907871513906110
