| L(s) = 1 | − 1.16·3-s + 2.71·5-s − 1.33·7-s − 1.63·9-s + 5.08·11-s − 0.388·13-s − 3.16·15-s + 0.121·17-s − 7.29·19-s + 1.55·21-s + 7.31·23-s + 2.36·25-s + 5.41·27-s − 8.97·29-s − 6.78·31-s − 5.93·33-s − 3.61·35-s − 0.924·37-s + 0.453·39-s + 2.63·41-s + 1.12·43-s − 4.44·45-s − 0.220·47-s − 5.22·49-s − 0.142·51-s − 7.43·53-s + 13.8·55-s + ⋯ |
| L(s) = 1 | − 0.674·3-s + 1.21·5-s − 0.503·7-s − 0.545·9-s + 1.53·11-s − 0.107·13-s − 0.817·15-s + 0.0295·17-s − 1.67·19-s + 0.339·21-s + 1.52·23-s + 0.472·25-s + 1.04·27-s − 1.66·29-s − 1.21·31-s − 1.03·33-s − 0.611·35-s − 0.151·37-s + 0.0726·39-s + 0.412·41-s + 0.171·43-s − 0.662·45-s − 0.0321·47-s − 0.746·49-s − 0.0199·51-s − 1.02·53-s + 1.86·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 1.16T + 3T^{2} \) |
| 5 | \( 1 - 2.71T + 5T^{2} \) |
| 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 - 5.08T + 11T^{2} \) |
| 13 | \( 1 + 0.388T + 13T^{2} \) |
| 17 | \( 1 - 0.121T + 17T^{2} \) |
| 19 | \( 1 + 7.29T + 19T^{2} \) |
| 23 | \( 1 - 7.31T + 23T^{2} \) |
| 29 | \( 1 + 8.97T + 29T^{2} \) |
| 31 | \( 1 + 6.78T + 31T^{2} \) |
| 37 | \( 1 + 0.924T + 37T^{2} \) |
| 41 | \( 1 - 2.63T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 0.220T + 47T^{2} \) |
| 53 | \( 1 + 7.43T + 53T^{2} \) |
| 59 | \( 1 - 1.34T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 - 0.702T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 7.50T + 73T^{2} \) |
| 79 | \( 1 - 5.63T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 1.99T + 97T^{2} \) |
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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
−7.57154555877559033067167216365, −6.57316459091718658796724151525, −6.41342923759717003404821985057, −5.69761584169683727881352258396, −5.04871977522162771334662416649, −4.07725743931094635119929298838, −3.22392987558319599013817512919, −2.17307187612081703668213704954, −1.37581048258648865129507225435, 0,
1.37581048258648865129507225435, 2.17307187612081703668213704954, 3.22392987558319599013817512919, 4.07725743931094635119929298838, 5.04871977522162771334662416649, 5.69761584169683727881352258396, 6.41342923759717003404821985057, 6.57316459091718658796724151525, 7.57154555877559033067167216365
