| L(s) = 1 | + 3.65·2-s + 3.36·3-s + 5.35·4-s + 5·5-s + 12.2·6-s − 1.36·7-s − 9.67·8-s − 15.7·9-s + 18.2·10-s − 11·11-s + 17.9·12-s − 71.5·13-s − 4.97·14-s + 16.8·15-s − 78.1·16-s + 55.9·17-s − 57.4·18-s − 19·19-s + 26.7·20-s − 4.57·21-s − 40.1·22-s + 125.·23-s − 32.5·24-s + 25·25-s − 261.·26-s − 143.·27-s − 7.29·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.646·3-s + 0.668·4-s + 0.447·5-s + 0.835·6-s − 0.0735·7-s − 0.427·8-s − 0.581·9-s + 0.577·10-s − 0.301·11-s + 0.432·12-s − 1.52·13-s − 0.0950·14-s + 0.289·15-s − 1.22·16-s + 0.797·17-s − 0.751·18-s − 0.229·19-s + 0.299·20-s − 0.0475·21-s − 0.389·22-s + 1.14·23-s − 0.276·24-s + 0.200·25-s − 1.97·26-s − 1.02·27-s − 0.0492·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 3.65T + 8T^{2} \) |
| 3 | \( 1 - 3.36T + 27T^{2} \) |
| 7 | \( 1 + 1.36T + 343T^{2} \) |
| 13 | \( 1 + 71.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 55.9T + 4.91e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 244.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 59.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 106.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 523.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 286.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 437.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 235.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 613.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 126.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 585.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 737.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 152.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 457.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−9.260840788392483496567291176208, −8.255989696767127234777703556556, −7.31634338137733242112003441316, −6.37846221426201238939695790797, −5.30520815472649854851203964959, −4.97790483636525875818517042612, −3.60120632058175240328274385186, −2.92686792655404057219774277390, −2.04949418716232816265662903279, 0,
2.04949418716232816265662903279, 2.92686792655404057219774277390, 3.60120632058175240328274385186, 4.97790483636525875818517042612, 5.30520815472649854851203964959, 6.37846221426201238939695790797, 7.31634338137733242112003441316, 8.255989696767127234777703556556, 9.260840788392483496567291176208
