| L(s) = 1 | + 1.84·3-s − 0.765·5-s − 7-s + 0.414·9-s − 1.53·11-s + 5.54·13-s − 1.41·15-s − 4.82·17-s + 2.93·19-s − 1.84·21-s + 3.41·23-s − 4.41·25-s − 4.77·27-s + 0.634·29-s + 6.82·31-s − 2.82·33-s + 0.765·35-s + 8.92·37-s + 10.2·39-s + 10.4·41-s + 3.69·43-s − 0.317·45-s + 4·47-s + 49-s − 8.92·51-s + 3.06·53-s + 1.17·55-s + ⋯ |
| L(s) = 1 | + 1.06·3-s − 0.342·5-s − 0.377·7-s + 0.138·9-s − 0.461·11-s + 1.53·13-s − 0.365·15-s − 1.17·17-s + 0.672·19-s − 0.403·21-s + 0.711·23-s − 0.882·25-s − 0.919·27-s + 0.117·29-s + 1.22·31-s − 0.492·33-s + 0.129·35-s + 1.46·37-s + 1.64·39-s + 1.63·41-s + 0.563·43-s − 0.0472·45-s + 0.583·47-s + 0.142·49-s − 1.24·51-s + 0.420·53-s + 0.157·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.458639445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.458639445\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 + 0.765T + 5T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 - 5.54T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 - 3.41T + 23T^{2} \) |
| 29 | \( 1 - 0.634T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 3.69T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 3.06T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 0.765T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 0.828T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 - 3.17T + 79T^{2} \) |
| 83 | \( 1 + 5.54T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.427389934770497965257435634458, −8.042097569909646958571649227736, −7.25854415326027994745850569907, −6.31110820068883500167203392498, −5.70278615150439147218185227676, −4.45400140891666091161612574545, −3.80559607003659133085286913051, −2.96749087197729437968140914994, −2.29564805135326596811201423809, −0.879003626013735253258174430769,
0.879003626013735253258174430769, 2.29564805135326596811201423809, 2.96749087197729437968140914994, 3.80559607003659133085286913051, 4.45400140891666091161612574545, 5.70278615150439147218185227676, 6.31110820068883500167203392498, 7.25854415326027994745850569907, 8.042097569909646958571649227736, 8.427389934770497965257435634458
