| L(s) = 1 | − 2.79·2-s + 3-s + 5.83·4-s − 1.67·5-s − 2.79·6-s + 7-s − 10.7·8-s + 9-s + 4.69·10-s + 5.83·12-s − 5.87·13-s − 2.79·14-s − 1.67·15-s + 18.3·16-s + 0.304·17-s − 2.79·18-s + 4.62·19-s − 9.78·20-s + 21-s − 1.38·23-s − 10.7·24-s − 2.18·25-s + 16.4·26-s + 27-s + 5.83·28-s − 3.96·29-s + 4.69·30-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 0.577·3-s + 2.91·4-s − 0.750·5-s − 1.14·6-s + 0.377·7-s − 3.79·8-s + 0.333·9-s + 1.48·10-s + 1.68·12-s − 1.62·13-s − 0.747·14-s − 0.433·15-s + 4.58·16-s + 0.0737·17-s − 0.659·18-s + 1.06·19-s − 2.18·20-s + 0.218·21-s − 0.288·23-s − 2.18·24-s − 0.437·25-s + 3.22·26-s + 0.192·27-s + 1.10·28-s − 0.736·29-s + 0.857·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6512754288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6512754288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 5 | \( 1 + 1.67T + 5T^{2} \) |
| 13 | \( 1 + 5.87T + 13T^{2} \) |
| 17 | \( 1 - 0.304T + 17T^{2} \) |
| 19 | \( 1 - 4.62T + 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 - 0.764T + 31T^{2} \) |
| 37 | \( 1 - 3.21T + 37T^{2} \) |
| 41 | \( 1 - 8.31T + 41T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 + 1.45T + 53T^{2} \) |
| 59 | \( 1 + 2.49T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 - 7.59T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 - 2.01T + 83T^{2} \) |
| 89 | \( 1 - 0.0843T + 89T^{2} \) |
| 97 | \( 1 - 1.03T + 97T^{2} \) |
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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
−8.960218748449808100330700144497, −7.991423471525045139522980957053, −7.66525821583645541739182261416, −7.25806464475339636727122138883, −6.22596026231619094038728876057, −5.09317138865691482081781962400, −3.70324060271243168589433072279, −2.70712787371251178358558212586, −1.94364303833359554735930392548, −0.64744399498650551871665087545,
0.64744399498650551871665087545, 1.94364303833359554735930392548, 2.70712787371251178358558212586, 3.70324060271243168589433072279, 5.09317138865691482081781962400, 6.22596026231619094038728876057, 7.25806464475339636727122138883, 7.66525821583645541739182261416, 7.991423471525045139522980957053, 8.960218748449808100330700144497
