| L(s) = 1 | − 5-s − 2·13-s + 6·23-s + 25-s + 6·29-s − 2·31-s − 2·37-s + 6·41-s − 4·43-s + 10·47-s − 7·49-s + 2·53-s − 8·59-s + 6·61-s + 2·65-s − 4·67-s + 16·71-s − 10·73-s − 14·79-s + 12·83-s + 2·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.554·13-s + 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s + 1.45·47-s − 49-s + 0.274·53-s − 1.04·59-s + 0.768·61-s + 0.248·65-s − 0.488·67-s + 1.89·71-s − 1.17·73-s − 1.57·79-s + 1.31·83-s + 0.211·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.90587891835188, −12.57452289772409, −12.21476359608504, −11.57018178920738, −11.30691951107962, −10.76214762556946, −10.26069099687374, −9.918597060908297, −9.205854348136056, −8.894460913072334, −8.456415432351417, −7.785373355125114, −7.449698214971081, −6.988372842314857, −6.460922675458417, −5.967824488106377, −5.284247094493156, −4.843921128005130, −4.468001857599859, −3.760782865451362, −3.279809338385207, −2.686527574865745, −2.216529329517179, −1.337674626729268, −0.7953891136586241, 0,
0.7953891136586241, 1.337674626729268, 2.216529329517179, 2.686527574865745, 3.279809338385207, 3.760782865451362, 4.468001857599859, 4.843921128005130, 5.284247094493156, 5.967824488106377, 6.460922675458417, 6.988372842314857, 7.449698214971081, 7.785373355125114, 8.456415432351417, 8.894460913072334, 9.205854348136056, 9.918597060908297, 10.26069099687374, 10.76214762556946, 11.30691951107962, 11.57018178920738, 12.21476359608504, 12.57452289772409, 12.90587891835188
