| L(s) = 1 | + 3.34e3·3-s − 5.21e4·5-s + 2.82e6·7-s − 3.13e6·9-s − 2.05e7·11-s + 1.90e8·13-s − 1.74e8·15-s + 1.64e9·17-s − 1.56e9·19-s + 9.44e9·21-s + 9.45e9·23-s − 2.78e10·25-s − 5.85e10·27-s + 3.69e10·29-s + 7.15e10·31-s − 6.89e10·33-s − 1.47e11·35-s + 1.03e12·37-s + 6.36e11·39-s + 1.64e12·41-s + 4.92e11·43-s + 1.63e11·45-s − 3.41e12·47-s + 3.21e12·49-s + 5.51e12·51-s − 6.79e12·53-s + 1.07e12·55-s + ⋯ |
| L(s) = 1 | + 0.883·3-s − 0.298·5-s + 1.29·7-s − 0.218·9-s − 0.318·11-s + 0.840·13-s − 0.263·15-s + 0.973·17-s − 0.401·19-s + 1.14·21-s + 0.578·23-s − 0.911·25-s − 1.07·27-s + 0.397·29-s + 0.467·31-s − 0.281·33-s − 0.386·35-s + 1.79·37-s + 0.742·39-s + 1.31·41-s + 0.276·43-s + 0.0652·45-s − 0.981·47-s + 0.677·49-s + 0.860·51-s − 0.794·53-s + 0.0950·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(3.501813890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.501813890\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 - 124 p^{3} T + p^{15} T^{2} \) |
| 5 | \( 1 + 10422 p T + p^{15} T^{2} \) |
| 7 | \( 1 - 403208 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 1871532 p T + p^{15} T^{2} \) |
| 13 | \( 1 - 14621026 p T + p^{15} T^{2} \) |
| 17 | \( 1 - 1646527986 T + p^{15} T^{2} \) |
| 19 | \( 1 + 1563257180 T + p^{15} T^{2} \) |
| 23 | \( 1 - 9451116072 T + p^{15} T^{2} \) |
| 29 | \( 1 - 36902568330 T + p^{15} T^{2} \) |
| 31 | \( 1 - 71588483552 T + p^{15} T^{2} \) |
| 37 | \( 1 - 1033652081554 T + p^{15} T^{2} \) |
| 41 | \( 1 - 1641974018202 T + p^{15} T^{2} \) |
| 43 | \( 1 - 492403109308 T + p^{15} T^{2} \) |
| 47 | \( 1 + 3410684952624 T + p^{15} T^{2} \) |
| 53 | \( 1 + 6797151655902 T + p^{15} T^{2} \) |
| 59 | \( 1 + 167099268060 p T + p^{15} T^{2} \) |
| 61 | \( 1 + 4931842626902 T + p^{15} T^{2} \) |
| 67 | \( 1 - 28837826625364 T + p^{15} T^{2} \) |
| 71 | \( 1 - 125050114914552 T + p^{15} T^{2} \) |
| 73 | \( 1 + 82171455513478 T + p^{15} T^{2} \) |
| 79 | \( 1 + 25413078694480 T + p^{15} T^{2} \) |
| 83 | \( 1 - 281736730890468 T + p^{15} T^{2} \) |
| 89 | \( 1 - 715618564776810 T + p^{15} T^{2} \) |
| 97 | \( 1 - 612786136081826 T + p^{15} T^{2} \) |
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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
−11.69492686423536833343056667306, −10.87331297369699508720297087020, −9.387830971244881215091244361673, −8.176665210235763338482847629267, −7.80660939569854791289806395274, −5.94592440467701426869344539467, −4.59534809984556872933090907663, −3.36018244780229563080309659926, −2.15379293314425097436531577478, −0.921057550725487266160513809352,
0.921057550725487266160513809352, 2.15379293314425097436531577478, 3.36018244780229563080309659926, 4.59534809984556872933090907663, 5.94592440467701426869344539467, 7.80660939569854791289806395274, 8.176665210235763338482847629267, 9.387830971244881215091244361673, 10.87331297369699508720297087020, 11.69492686423536833343056667306
