| L(s) = 1 | + 0.280·3-s + 1.39·5-s + 1.54·7-s − 2.92·9-s − 1.91·13-s + 0.392·15-s + 0.813·17-s + 19-s + 0.434·21-s + 2.02·23-s − 3.04·25-s − 1.66·27-s − 8.17·29-s + 9.56·31-s + 2.16·35-s − 10.7·37-s − 0.536·39-s + 1.31·41-s − 1.86·43-s − 4.07·45-s + 3.67·47-s − 4.60·49-s + 0.228·51-s + 8.16·53-s + 0.280·57-s − 6.79·59-s − 0.315·61-s + ⋯ |
| L(s) = 1 | + 0.162·3-s + 0.624·5-s + 0.585·7-s − 0.973·9-s − 0.530·13-s + 0.101·15-s + 0.197·17-s + 0.229·19-s + 0.0948·21-s + 0.422·23-s − 0.609·25-s − 0.319·27-s − 1.51·29-s + 1.71·31-s + 0.365·35-s − 1.76·37-s − 0.0859·39-s + 0.206·41-s − 0.283·43-s − 0.608·45-s + 0.536·47-s − 0.657·49-s + 0.0319·51-s + 1.12·53-s + 0.0371·57-s − 0.884·59-s − 0.0403·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.280T + 3T^{2} \) |
| 5 | \( 1 - 1.39T + 5T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 13 | \( 1 + 1.91T + 13T^{2} \) |
| 17 | \( 1 - 0.813T + 17T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 1.31T + 41T^{2} \) |
| 43 | \( 1 + 1.86T + 43T^{2} \) |
| 47 | \( 1 - 3.67T + 47T^{2} \) |
| 53 | \( 1 - 8.16T + 53T^{2} \) |
| 59 | \( 1 + 6.79T + 59T^{2} \) |
| 61 | \( 1 + 0.315T + 61T^{2} \) |
| 67 | \( 1 - 3.73T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 8.84T + 83T^{2} \) |
| 89 | \( 1 - 7.43T + 89T^{2} \) |
| 97 | \( 1 + 5.01T + 97T^{2} \) |
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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
−7.50083944522449741560720828829, −6.67268656442034065596262081909, −5.87779051854545043041349553951, −5.37920289559487954175789285507, −4.74559701804946652479046988064, −3.77089863146973720088184973707, −2.93432667291028586664026642150, −2.21263652390491905274461375537, −1.38285583806832167209288690461, 0,
1.38285583806832167209288690461, 2.21263652390491905274461375537, 2.93432667291028586664026642150, 3.77089863146973720088184973707, 4.74559701804946652479046988064, 5.37920289559487954175789285507, 5.87779051854545043041349553951, 6.67268656442034065596262081909, 7.50083944522449741560720828829
