| L(s) = 1 | + 2.45·2-s + 1.56·3-s + 4.03·4-s + 3.83·6-s − 4.50·7-s + 4.99·8-s − 0.563·9-s + 2.19·11-s + 6.29·12-s − 3.75·13-s − 11.0·14-s + 4.19·16-s − 0.665·17-s − 1.38·18-s − 7.03·21-s + 5.40·22-s + 0.488·23-s + 7.79·24-s − 9.23·26-s − 5.56·27-s − 18.1·28-s − 3.59·29-s − 6.83·31-s + 0.326·32-s + 3.43·33-s − 1.63·34-s − 2.27·36-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 0.901·3-s + 2.01·4-s + 1.56·6-s − 1.70·7-s + 1.76·8-s − 0.187·9-s + 0.662·11-s + 1.81·12-s − 1.04·13-s − 2.95·14-s + 1.04·16-s − 0.161·17-s − 0.326·18-s − 1.53·21-s + 1.15·22-s + 0.101·23-s + 1.59·24-s − 1.81·26-s − 1.07·27-s − 3.43·28-s − 0.667·29-s − 1.22·31-s + 0.0576·32-s + 0.597·33-s − 0.280·34-s − 0.378·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 4.50T + 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 + 0.665T + 17T^{2} \) |
| 23 | \( 1 - 0.488T + 23T^{2} \) |
| 29 | \( 1 + 3.59T + 29T^{2} \) |
| 31 | \( 1 + 6.83T + 31T^{2} \) |
| 37 | \( 1 + 3.01T + 37T^{2} \) |
| 41 | \( 1 + 0.0724T + 41T^{2} \) |
| 43 | \( 1 + 0.420T + 43T^{2} \) |
| 47 | \( 1 + 5.02T + 47T^{2} \) |
| 53 | \( 1 - 2.61T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 7.06T + 61T^{2} \) |
| 67 | \( 1 - 5.72T + 67T^{2} \) |
| 71 | \( 1 + 6.97T + 71T^{2} \) |
| 73 | \( 1 - 2.95T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 - 4.38T + 97T^{2} \) |
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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
−7.10536943123783990653070006068, −6.61569007421177222772309212990, −5.91904033513564998801896689206, −5.33595921265431120967694679507, −4.42178988759466667236073036611, −3.59933211270721734681109248698, −3.33972956835850875894087021456, −2.60014833612717651113545749361, −1.92031339426973599159852261100, 0,
1.92031339426973599159852261100, 2.60014833612717651113545749361, 3.33972956835850875894087021456, 3.59933211270721734681109248698, 4.42178988759466667236073036611, 5.33595921265431120967694679507, 5.91904033513564998801896689206, 6.61569007421177222772309212990, 7.10536943123783990653070006068
