L(s) = 1 | − 696. i·3-s + 7.35e4i·7-s − 3.07e5·9-s − 3.83e5·11-s − 1.15e6i·13-s − 6.51e6i·17-s + 1.39e7·19-s + 5.12e7·21-s − 1.37e6i·23-s + 9.07e7i·27-s + 7.46e7·29-s + 1.32e7·31-s + 2.66e8i·33-s − 1.67e7i·37-s − 8.06e8·39-s + ⋯ |
L(s) = 1 | − 1.65i·3-s + 1.65i·7-s − 1.73·9-s − 0.717·11-s − 0.865i·13-s − 1.11i·17-s + 1.29·19-s + 2.73·21-s − 0.0445i·23-s + 1.21i·27-s + 0.675·29-s + 0.0829·31-s + 1.18i·33-s − 0.0396i·37-s − 1.43·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.8565789232\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8565789232\) |
\(L(\frac{13}{2})\) |
|
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 | \( 1 + 696. iT - 1.77e5T^{2} \) |
| 7 | \( 1 - 7.35e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 3.83e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.15e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 6.51e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 1.39e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.37e6iT - 9.52e14T^{2} \) |
| 29 | \( 1 - 7.46e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.32e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 1.67e7iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 1.03e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.93e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 + 1.16e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 4.44e8iT - 9.26e18T^{2} \) |
| 59 | \( 1 - 1.28e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 7.96e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 6.89e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 1.12e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.34e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 5.36e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 6.31e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + 9.62e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.37e10iT - 7.15e21T^{2} \) |
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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
−9.728093769929389519040964745972, −8.618360722088657985562411907605, −7.85821688012521427417903440598, −6.98334359644548105306869377393, −5.78044247621180367837471060008, −5.26073878123809510787370663016, −2.86412336350785449634464107283, −2.48392457450707005390787738423, −1.18838479210467286673653522501, −0.18045297029743166718863332116,
1.17584941275648228869495108951, 2.98027248315862085449081067739, 4.01597623716675698287438478954, 4.50949493372011150233607435633, 5.69121945706175746017990868829, 7.09593921504717769853294609309, 8.174678177463881323392332432430, 9.396511663214024326015265417465, 10.14926358557688431290826411106, 10.72075995258545223584993685455