| L(s) = 1 | + 2.75e3·3-s − 1.08e5·5-s + 2.79e6·7-s − 6.74e6·9-s + 3.90e7·11-s − 1.19e8·13-s − 3.00e8·15-s − 1.44e8·17-s + 6.63e9·19-s + 7.71e9·21-s + 1.17e10·23-s − 1.86e10·25-s − 5.81e10·27-s + 1.35e9·29-s + 8.88e10·31-s + 1.07e11·33-s − 3.04e11·35-s + 2.82e11·37-s − 3.30e11·39-s + 1.12e12·41-s − 2.25e10·43-s + 7.34e11·45-s + 5.31e12·47-s + 3.07e12·49-s − 3.98e11·51-s + 1.34e13·53-s − 4.25e12·55-s + ⋯ |
| L(s) = 1 | + 0.727·3-s − 0.623·5-s + 1.28·7-s − 0.470·9-s + 0.604·11-s − 0.530·13-s − 0.453·15-s − 0.0854·17-s + 1.70·19-s + 0.934·21-s + 0.719·23-s − 0.611·25-s − 1.07·27-s + 0.0146·29-s + 0.580·31-s + 0.439·33-s − 0.799·35-s + 0.488·37-s − 0.386·39-s + 0.899·41-s − 0.0126·43-s + 0.293·45-s + 1.52·47-s + 0.647·49-s − 0.0622·51-s + 1.57·53-s − 0.376·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(2.797479442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.797479442\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.75e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 1.08e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 2.79e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 3.90e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 1.19e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.44e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 6.63e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 1.17e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 1.35e9T + 8.62e21T^{2} \) |
| 31 | \( 1 - 8.88e10T + 2.34e22T^{2} \) |
| 37 | \( 1 - 2.82e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 1.12e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 2.25e10T + 3.17e24T^{2} \) |
| 47 | \( 1 - 5.31e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 1.34e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 1.47e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 3.94e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 8.22e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 7.97e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.03e14T + 8.90e27T^{2} \) |
| 79 | \( 1 - 2.49e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 3.67e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + 7.06e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.40e15T + 6.33e29T^{2} \) |
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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
−13.79544839144540533743340700283, −11.97198705290252770163070368036, −11.22982443485285278045887242993, −9.432543564060738033358214377674, −8.255683349143794865938913828518, −7.37981489367896883650033060056, −5.32962996102068298878373654536, −3.92873588961967973087349051123, −2.50833515403247027245766666923, −0.973772409920768027263188547750,
0.973772409920768027263188547750, 2.50833515403247027245766666923, 3.92873588961967973087349051123, 5.32962996102068298878373654536, 7.37981489367896883650033060056, 8.255683349143794865938913828518, 9.432543564060738033358214377674, 11.22982443485285278045887242993, 11.97198705290252770163070368036, 13.79544839144540533743340700283
