| L(s) = 1 | − 2-s + 4-s − 0.347·5-s − 8-s + 0.347·10-s + 1.53·13-s + 16-s + 1.87·17-s − 0.347·20-s − 0.879·25-s − 1.53·26-s − 1.53·29-s − 32-s − 1.87·34-s − 1.87·37-s + 0.347·40-s + 41-s + 49-s + 0.879·50-s + 1.53·52-s + 53-s + 1.53·58-s + 1.53·61-s + 64-s − 0.532·65-s + 1.87·68-s + 0.347·73-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 0.347·5-s − 8-s + 0.347·10-s + 1.53·13-s + 16-s + 1.87·17-s − 0.347·20-s − 0.879·25-s − 1.53·26-s − 1.53·29-s − 32-s − 1.87·34-s − 1.87·37-s + 0.347·40-s + 41-s + 49-s + 0.879·50-s + 1.53·52-s + 53-s + 1.53·58-s + 1.53·61-s + 64-s − 0.532·65-s + 1.87·68-s + 0.347·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8216191257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8216191257\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.347T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.53T + T^{2} \) |
| 17 | \( 1 - 1.87T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.53T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.87T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.53T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.347T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.87T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.887218297913576718241052012263, −8.211459702143612243996295641896, −7.60530292378119634323509758882, −6.93298805025348307662276439779, −5.86156311357571870920210838371, −5.50748082439024894203809636969, −3.80421022166479137950161263206, −3.41014288895231580201693849629, −2.03422997130692547851069777448, −1.00108169768533622710915083698,
1.00108169768533622710915083698, 2.03422997130692547851069777448, 3.41014288895231580201693849629, 3.80421022166479137950161263206, 5.50748082439024894203809636969, 5.86156311357571870920210838371, 6.93298805025348307662276439779, 7.60530292378119634323509758882, 8.211459702143612243996295641896, 8.887218297913576718241052012263
