| L(s) = 1 | − 2.21·3-s − 3.70·5-s − 3.08·7-s + 1.90·9-s − 5.39·11-s − 1.54·13-s + 8.19·15-s + 17-s + 5.15·19-s + 6.83·21-s + 6.37·23-s + 8.71·25-s + 2.43·27-s + 4.84·29-s − 8.51·31-s + 11.9·33-s + 11.4·35-s + 2.87·37-s + 3.42·39-s − 6.72·41-s + 11.4·43-s − 7.03·45-s − 6.97·47-s + 2.54·49-s − 2.21·51-s + 4.82·53-s + 19.9·55-s + ⋯ |
| L(s) = 1 | − 1.27·3-s − 1.65·5-s − 1.16·7-s + 0.633·9-s − 1.62·11-s − 0.429·13-s + 2.11·15-s + 0.242·17-s + 1.18·19-s + 1.49·21-s + 1.32·23-s + 1.74·25-s + 0.468·27-s + 0.898·29-s − 1.53·31-s + 2.07·33-s + 1.93·35-s + 0.471·37-s + 0.548·39-s − 1.05·41-s + 1.74·43-s − 1.04·45-s − 1.01·47-s + 0.363·49-s − 0.309·51-s + 0.662·53-s + 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2.21T + 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 + 5.39T + 11T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 19 | \( 1 - 5.15T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 - 4.84T + 29T^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 - 2.87T + 37T^{2} \) |
| 41 | \( 1 + 6.72T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 6.97T + 47T^{2} \) |
| 53 | \( 1 - 4.82T + 53T^{2} \) |
| 61 | \( 1 - 1.29T + 61T^{2} \) |
| 67 | \( 1 + 1.85T + 67T^{2} \) |
| 71 | \( 1 + 3.42T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 6.18T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 3.28T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−7.84908969214528088983121713899, −7.25476138253708030654807877014, −6.78332452760710257051681608557, −5.67154316828390886583537263670, −5.18013195505088899019344119233, −4.43466471529849820841776697927, −3.31709388542210746182364078753, −2.85958783325575285034869470073, −0.75710249449547025861077381000, 0,
0.75710249449547025861077381000, 2.85958783325575285034869470073, 3.31709388542210746182364078753, 4.43466471529849820841776697927, 5.18013195505088899019344119233, 5.67154316828390886583537263670, 6.78332452760710257051681608557, 7.25476138253708030654807877014, 7.84908969214528088983121713899
