| L(s) = 1 | − 1.91·3-s + 2.91·5-s + 0.656·9-s − 0.656·11-s − 2.91·13-s − 5.56·15-s − 17-s + 2.82·19-s + 5.13·23-s + 3.48·25-s + 4.48·27-s − 8.13·29-s − 0.656·31-s + 1.25·33-s − 9.88·37-s + 5.56·39-s − 4.25·41-s − 5.51·43-s + 1.91·45-s − 6.31·47-s + 1.91·51-s + 2.48·53-s − 1.91·55-s − 5.40·57-s + 12.5·59-s + 9.36·61-s − 8.48·65-s + ⋯ |
| L(s) = 1 | − 1.10·3-s + 1.30·5-s + 0.218·9-s − 0.197·11-s − 0.807·13-s − 1.43·15-s − 0.242·17-s + 0.647·19-s + 1.07·23-s + 0.696·25-s + 0.862·27-s − 1.51·29-s − 0.117·31-s + 0.218·33-s − 1.62·37-s + 0.891·39-s − 0.664·41-s − 0.840·43-s + 0.285·45-s − 0.920·47-s + 0.267·51-s + 0.340·53-s − 0.257·55-s − 0.715·57-s + 1.63·59-s + 1.19·61-s − 1.05·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 - 2.91T + 5T^{2} \) |
| 11 | \( 1 + 0.656T + 11T^{2} \) |
| 13 | \( 1 + 2.91T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 5.13T + 23T^{2} \) |
| 29 | \( 1 + 8.13T + 29T^{2} \) |
| 31 | \( 1 + 0.656T + 31T^{2} \) |
| 37 | \( 1 + 9.88T + 37T^{2} \) |
| 41 | \( 1 + 4.25T + 41T^{2} \) |
| 43 | \( 1 + 5.51T + 43T^{2} \) |
| 47 | \( 1 + 6.31T + 47T^{2} \) |
| 53 | \( 1 - 2.48T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 8.02T + 71T^{2} \) |
| 73 | \( 1 + 0.538T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 2.05T + 97T^{2} \) |
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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
−8.347667541696685517935708811181, −7.12069560260855003272034695560, −6.77965803868291140379491253542, −5.75358250584647100631661542187, −5.35532550341480868254742459473, −4.83319136692684225290131912205, −3.43998766125806114590226618493, −2.38888026575971045034901066378, −1.42986931812923695667678159344, 0,
1.42986931812923695667678159344, 2.38888026575971045034901066378, 3.43998766125806114590226618493, 4.83319136692684225290131912205, 5.35532550341480868254742459473, 5.75358250584647100631661542187, 6.77965803868291140379491253542, 7.12069560260855003272034695560, 8.347667541696685517935708811181
