| L(s) = 1 | − 1.77·3-s + 3.93·5-s − 3.71·7-s + 0.167·9-s + 3.12·11-s − 5.91·13-s − 7.00·15-s − 0.159·17-s − 2.26·19-s + 6.60·21-s − 9.26·23-s + 10.5·25-s + 5.04·27-s + 7.78·29-s + 5.11·31-s − 5.56·33-s − 14.6·35-s + 11.0·37-s + 10.5·39-s − 0.215·41-s + 1.14·43-s + 0.660·45-s + 0.412·47-s + 6.77·49-s + 0.284·51-s − 3.01·53-s + 12.3·55-s + ⋯ |
| L(s) = 1 | − 1.02·3-s + 1.76·5-s − 1.40·7-s + 0.0559·9-s + 0.943·11-s − 1.64·13-s − 1.80·15-s − 0.0387·17-s − 0.520·19-s + 1.44·21-s − 1.93·23-s + 2.10·25-s + 0.970·27-s + 1.44·29-s + 0.919·31-s − 0.969·33-s − 2.47·35-s + 1.81·37-s + 1.68·39-s − 0.0337·41-s + 0.174·43-s + 0.0985·45-s + 0.0601·47-s + 0.967·49-s + 0.0398·51-s − 0.414·53-s + 1.66·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 1.77T + 3T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 + 0.159T + 17T^{2} \) |
| 19 | \( 1 + 2.26T + 19T^{2} \) |
| 23 | \( 1 + 9.26T + 23T^{2} \) |
| 29 | \( 1 - 7.78T + 29T^{2} \) |
| 31 | \( 1 - 5.11T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 0.215T + 41T^{2} \) |
| 43 | \( 1 - 1.14T + 43T^{2} \) |
| 47 | \( 1 - 0.412T + 47T^{2} \) |
| 53 | \( 1 + 3.01T + 53T^{2} \) |
| 59 | \( 1 - 8.18T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 4.37T + 67T^{2} \) |
| 71 | \( 1 + 8.95T + 71T^{2} \) |
| 73 | \( 1 + 7.82T + 73T^{2} \) |
| 79 | \( 1 - 5.80T + 79T^{2} \) |
| 83 | \( 1 + 8.05T + 83T^{2} \) |
| 89 | \( 1 - 1.45T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.07782901374068971323702849290, −6.39184219233325895170527900497, −6.25611458496089062465153686607, −5.64902389378669240679873498914, −4.83792131138359162611477210670, −4.09587976661852344741742911683, −2.72098869250398447709218019683, −2.41477260811337503085690223992, −1.13146433121303823372975467216, 0,
1.13146433121303823372975467216, 2.41477260811337503085690223992, 2.72098869250398447709218019683, 4.09587976661852344741742911683, 4.83792131138359162611477210670, 5.64902389378669240679873498914, 6.25611458496089062465153686607, 6.39184219233325895170527900497, 7.07782901374068971323702849290
