L(s) = 1 | + 3-s + 2.60·5-s + 3.58·7-s + 9-s + 4.72·11-s − 0.942·13-s + 2.60·15-s − 3.64·17-s + 2.69·19-s + 3.58·21-s + 0.381·23-s + 1.76·25-s + 27-s + 4.31·29-s − 0.400·31-s + 4.72·33-s + 9.31·35-s + 0.407·37-s − 0.942·39-s + 0.759·41-s − 4.47·43-s + 2.60·45-s − 13.2·47-s + 5.82·49-s − 3.64·51-s + 4.41·53-s + 12.2·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.16·5-s + 1.35·7-s + 0.333·9-s + 1.42·11-s − 0.261·13-s + 0.671·15-s − 0.884·17-s + 0.617·19-s + 0.781·21-s + 0.0795·23-s + 0.353·25-s + 0.192·27-s + 0.800·29-s − 0.0719·31-s + 0.822·33-s + 1.57·35-s + 0.0669·37-s − 0.150·39-s + 0.118·41-s − 0.682·43-s + 0.387·45-s − 1.92·47-s + 0.832·49-s − 0.510·51-s + 0.606·53-s + 1.65·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.520352841\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.520352841\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 - 3.58T + 7T^{2} \) |
| 11 | \( 1 - 4.72T + 11T^{2} \) |
| 13 | \( 1 + 0.942T + 13T^{2} \) |
| 17 | \( 1 + 3.64T + 17T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 23 | \( 1 - 0.381T + 23T^{2} \) |
| 29 | \( 1 - 4.31T + 29T^{2} \) |
| 31 | \( 1 + 0.400T + 31T^{2} \) |
| 37 | \( 1 - 0.407T + 37T^{2} \) |
| 41 | \( 1 - 0.759T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 4.41T + 53T^{2} \) |
| 59 | \( 1 - 5.47T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 4.38T + 67T^{2} \) |
| 71 | \( 1 - 1.17T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 8.57T + 79T^{2} \) |
| 83 | \( 1 + 0.551T + 83T^{2} \) |
| 89 | \( 1 - 0.764T + 89T^{2} \) |
| 97 | \( 1 + 4.11T + 97T^{2} \) |
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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
−8.040515106553373189541289125994, −6.98189068741777118209290693663, −6.62152611468076432096361288806, −5.70862502866542393286411995801, −4.97266737421881251826002294945, −4.38157248671722446130885583509, −3.50674997877368967736214908113, −2.43301935993732376572678658112, −1.79033849915491980871183302615, −1.15224419493487414603613706891,
1.15224419493487414603613706891, 1.79033849915491980871183302615, 2.43301935993732376572678658112, 3.50674997877368967736214908113, 4.38157248671722446130885583509, 4.97266737421881251826002294945, 5.70862502866542393286411995801, 6.62152611468076432096361288806, 6.98189068741777118209290693663, 8.040515106553373189541289125994