| L(s) = 1 | + 0.755·3-s + 172.·7-s − 242.·9-s + 391.·11-s − 149.·13-s − 1.18e3·17-s + 685.·19-s + 130.·21-s − 996.·23-s − 366.·27-s − 8.76e3·29-s + 9.52e3·31-s + 295.·33-s − 1.02e4·37-s − 112.·39-s + 32.6·41-s − 1.03e4·43-s − 1.69e4·47-s + 1.29e4·49-s − 897.·51-s + 2.22e4·53-s + 517.·57-s + 4.42e4·59-s − 2.17e4·61-s − 4.18e4·63-s − 2.07e4·67-s − 752.·69-s + ⋯ |
| L(s) = 1 | + 0.0484·3-s + 1.33·7-s − 0.997·9-s + 0.975·11-s − 0.244·13-s − 0.996·17-s + 0.435·19-s + 0.0644·21-s − 0.392·23-s − 0.0968·27-s − 1.93·29-s + 1.78·31-s + 0.0472·33-s − 1.22·37-s − 0.0118·39-s + 0.00303·41-s − 0.851·43-s − 1.11·47-s + 0.768·49-s − 0.0483·51-s + 1.08·53-s + 0.0211·57-s + 1.65·59-s − 0.748·61-s − 1.32·63-s − 0.563·67-s − 0.0190·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.755T + 243T^{2} \) |
| 7 | \( 1 - 172.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 391.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 149.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 685.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 996.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.02e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 32.6T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.24e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.22e4T + 8.58e9T^{2} \) |
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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
−8.868787216475197602802351027336, −8.398077045097479432543814897021, −7.44620950201920194492122060525, −6.46322592483290446341540496049, −5.46076159622524194712600266304, −4.63857340623051327807796759207, −3.61717518152042065233602931489, −2.31458145004354685719862793397, −1.39720377545198584190087226031, 0,
1.39720377545198584190087226031, 2.31458145004354685719862793397, 3.61717518152042065233602931489, 4.63857340623051327807796759207, 5.46076159622524194712600266304, 6.46322592483290446341540496049, 7.44620950201920194492122060525, 8.398077045097479432543814897021, 8.868787216475197602802351027336
