L(s) = 1 | + 18i·3-s + 242i·7-s − 81·9-s + 656·11-s + 206i·13-s + 1.69e3i·17-s + 1.36e3·19-s − 4.35e3·21-s − 2.19e3i·23-s + 2.91e3i·27-s + 2.21e3·29-s − 1.70e3·31-s + 1.18e4i·33-s − 846i·37-s − 3.70e3·39-s + ⋯ |
L(s) = 1 | + 1.15i·3-s + 1.86i·7-s − 0.333·9-s + 1.63·11-s + 0.338i·13-s + 1.41i·17-s + 0.866·19-s − 2.15·21-s − 0.866i·23-s + 0.769i·27-s + 0.489·29-s − 0.317·31-s + 1.88i·33-s − 0.101i·37-s − 0.390·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.148794558\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.148794558\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 - 18iT - 243T^{2} \) |
| 7 | \( 1 - 242iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 656T + 1.61e5T^{2} \) |
| 13 | \( 1 - 206iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.69e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.19e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 846iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.81e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.05e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.20e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.25e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 8.66e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.46e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.75e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 948T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.31e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.87e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.04e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.62e4iT - 8.58e9T^{2} \) |
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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
−11.96055177592254939183594249237, −11.03925985308613112581005577329, −9.813239160732917343498780638489, −9.112415629263569486379147222845, −8.453092380269791844209022712527, −6.57724988554881623945761834069, −5.63093695567149132903309164600, −4.45329224467765963188390326017, −3.35065002446976883109434462910, −1.76865738407158621027249172114,
0.74123649422388518967884185223, 1.38675957037790111408376744312, 3.36030051429201939748214060044, 4.56311255729882786575800860023, 6.31107529601396488870191056785, 7.24781720247463542924883917758, 7.58603119174489860603208742333, 9.205397388826894397048941373039, 10.17049105441251841046832650518, 11.41557916868221746021688632943