| L(s) = 1 | + 3-s − 2.93·5-s − 2.93·7-s + 9-s + 1.68·11-s + 5.57·13-s − 2.93·15-s + 0.319·17-s + 4.93·19-s − 2.93·21-s + 0.939·23-s + 3.63·25-s + 27-s + 2.31·29-s − 7.89·31-s + 1.68·33-s + 8.63·35-s − 5.89·37-s + 5.57·39-s − 41-s − 9.77·43-s − 2.93·45-s − 4.31·47-s + 1.63·49-s + 0.319·51-s + 4·53-s − 4.93·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.31·5-s − 1.11·7-s + 0.333·9-s + 0.506·11-s + 1.54·13-s − 0.758·15-s + 0.0775·17-s + 1.13·19-s − 0.641·21-s + 0.195·23-s + 0.727·25-s + 0.192·27-s + 0.430·29-s − 1.41·31-s + 0.292·33-s + 1.46·35-s − 0.969·37-s + 0.893·39-s − 0.156·41-s − 1.49·43-s − 0.438·45-s − 0.630·47-s + 0.234·49-s + 0.0447·51-s + 0.549·53-s − 0.666·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.668345857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668345857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 + 2.93T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 - 5.57T + 13T^{2} \) |
| 17 | \( 1 - 0.319T + 17T^{2} \) |
| 19 | \( 1 - 4.93T + 19T^{2} \) |
| 23 | \( 1 - 0.939T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 + 5.89T + 37T^{2} \) |
| 43 | \( 1 + 9.77T + 43T^{2} \) |
| 47 | \( 1 + 4.31T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 2.12T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 - 3.27T + 79T^{2} \) |
| 83 | \( 1 - 2.42T + 83T^{2} \) |
| 89 | \( 1 + 9.75T + 89T^{2} \) |
| 97 | \( 1 - 6.30T + 97T^{2} \) |
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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
−7.898631125190020052654621536403, −7.11373353404973816140501885638, −6.71130145489702363567240089526, −5.85339389220119961467379927291, −4.95146399699658053986864674273, −3.86719401284642094718902904242, −3.54706145592250692227909237429, −3.11713202787567707424491918746, −1.70641813811007705627118891972, −0.63493350861160297208634948767,
0.63493350861160297208634948767, 1.70641813811007705627118891972, 3.11713202787567707424491918746, 3.54706145592250692227909237429, 3.86719401284642094718902904242, 4.95146399699658053986864674273, 5.85339389220119961467379927291, 6.71130145489702363567240089526, 7.11373353404973816140501885638, 7.898631125190020052654621536403
