| L(s) = 1 | − 3-s + 1.83·5-s + 3.50·7-s + 9-s + 3.38·11-s + 3.23·13-s − 1.83·15-s + 7.94·19-s − 3.50·21-s − 2.60·23-s − 1.61·25-s − 27-s + 5.52·29-s + 10.4·31-s − 3.38·33-s + 6.44·35-s + 9.43·37-s − 3.23·39-s − 9.70·41-s − 4.03·43-s + 1.83·45-s + 1.91·47-s + 5.26·49-s + 6.44·53-s + 6.21·55-s − 7.94·57-s − 12.7·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.822·5-s + 1.32·7-s + 0.333·9-s + 1.01·11-s + 0.898·13-s − 0.474·15-s + 1.82·19-s − 0.764·21-s − 0.544·23-s − 0.323·25-s − 0.192·27-s + 1.02·29-s + 1.87·31-s − 0.588·33-s + 1.08·35-s + 1.55·37-s − 0.518·39-s − 1.51·41-s − 0.614·43-s + 0.274·45-s + 0.279·47-s + 0.752·49-s + 0.885·53-s + 0.838·55-s − 1.05·57-s − 1.65·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.092148166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.092148166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 1.83T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 19 | \( 1 - 7.94T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 - 5.52T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 9.43T + 37T^{2} \) |
| 41 | \( 1 + 9.70T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 - 1.91T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 1.66T + 61T^{2} \) |
| 67 | \( 1 - 6.83T + 67T^{2} \) |
| 71 | \( 1 + 6.33T + 71T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 + 5.34T + 79T^{2} \) |
| 83 | \( 1 + 3.98T + 83T^{2} \) |
| 89 | \( 1 - 8.16T + 89T^{2} \) |
| 97 | \( 1 + 2.07T + 97T^{2} \) |
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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
−8.037774029052206016156695587007, −7.20244224177136890893084569978, −6.32583613087903963100752559705, −5.95954148979579042908177047967, −5.10550687356992920338590656211, −4.56516944780872129071715661502, −3.69750229723344114790565693849, −2.61154020429328550418271553324, −1.42267610430723820838074458808, −1.14789214875736982151468985051,
1.14789214875736982151468985051, 1.42267610430723820838074458808, 2.61154020429328550418271553324, 3.69750229723344114790565693849, 4.56516944780872129071715661502, 5.10550687356992920338590656211, 5.95954148979579042908177047967, 6.32583613087903963100752559705, 7.20244224177136890893084569978, 8.037774029052206016156695587007
