| L(s) = 1 | + 2-s − 2.80·3-s + 4-s − 0.947·5-s − 2.80·6-s + 2.44·7-s + 8-s + 4.86·9-s − 0.947·10-s + 0.609·11-s − 2.80·12-s + 3.92·13-s + 2.44·14-s + 2.65·15-s + 16-s − 1.10·17-s + 4.86·18-s − 6.74·19-s − 0.947·20-s − 6.87·21-s + 0.609·22-s + 23-s − 2.80·24-s − 4.10·25-s + 3.92·26-s − 5.23·27-s + 2.44·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.61·3-s + 0.5·4-s − 0.423·5-s − 1.14·6-s + 0.925·7-s + 0.353·8-s + 1.62·9-s − 0.299·10-s + 0.183·11-s − 0.809·12-s + 1.08·13-s + 0.654·14-s + 0.686·15-s + 0.250·16-s − 0.269·17-s + 1.14·18-s − 1.54·19-s − 0.211·20-s − 1.49·21-s + 0.130·22-s + 0.208·23-s − 0.572·24-s − 0.820·25-s + 0.769·26-s − 1.00·27-s + 0.462·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 0.947T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 0.609T + 11T^{2} \) |
| 13 | \( 1 - 3.92T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 29 | \( 1 + 4.15T + 29T^{2} \) |
| 31 | \( 1 + 6.83T + 31T^{2} \) |
| 37 | \( 1 + 1.29T + 37T^{2} \) |
| 41 | \( 1 - 0.848T + 41T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 + 2.36T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 6.86T + 67T^{2} \) |
| 71 | \( 1 - 6.33T + 71T^{2} \) |
| 73 | \( 1 + 3.30T + 73T^{2} \) |
| 79 | \( 1 + 2.00T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.51595350413202454264374956424, −6.73621302066781355899511705624, −6.14718055055235549929578791659, −5.58020042692687096091370247518, −4.89876027271531970873458166362, −4.17840486262629774562025522506, −3.70391700755194706783125770646, −2.13040197054249708208052790449, −1.30636654842061360204049909134, 0,
1.30636654842061360204049909134, 2.13040197054249708208052790449, 3.70391700755194706783125770646, 4.17840486262629774562025522506, 4.89876027271531970873458166362, 5.58020042692687096091370247518, 6.14718055055235549929578791659, 6.73621302066781355899511705624, 7.51595350413202454264374956424
