L(s) = 1 | + 15.8·3-s − 40.8i·5-s + 40.7i·7-s + 170.·9-s + 42.9·11-s − 186. i·13-s − 647. i·15-s − 157.·17-s + 278.·19-s + 646. i·21-s + 249. i·23-s − 1.04e3·25-s + 1.41e3·27-s + 416. i·29-s − 1.08e3i·31-s + ⋯ |
L(s) = 1 | + 1.76·3-s − 1.63i·5-s + 0.832i·7-s + 2.10·9-s + 0.354·11-s − 1.10i·13-s − 2.87i·15-s − 0.544·17-s + 0.770·19-s + 1.46i·21-s + 0.472i·23-s − 1.66·25-s + 1.94·27-s + 0.495i·29-s − 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.99937 - 1.24238i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.99937 - 1.24238i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 15.8T + 81T^{2} \) |
| 5 | \( 1 + 40.8iT - 625T^{2} \) |
| 7 | \( 1 - 40.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 42.9T + 1.46e4T^{2} \) |
| 13 | \( 1 + 186. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 157.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 278.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 249. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 416. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.08e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 742. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.19e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.25e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.79e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.87e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 5.25e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 1.36e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 316.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 8.96e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 9.72e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.46e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.06e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + 6.00e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 7.30e3T + 8.85e7T^{2} \) |
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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
−12.90857437758921897355059069722, −11.82144187173997871797289841296, −9.861374680377803837079769338096, −9.059398192696331364571492772233, −8.488627377776261837286354608424, −7.59796394089058595967417388784, −5.54556086301526975909111724235, −4.23043907873983868681884266792, −2.79119059054889174194049203806, −1.32054362175796475633973491835,
2.00918425621220278403921802539, 3.20906605148495097958932794686, 4.11007946138113761234695462528, 6.79611096446234452176820716885, 7.22111101788875447112530917327, 8.491984951975585990674856151863, 9.634171898472767290775539318614, 10.44371685875061902638459859093, 11.59908790157788077759379623224, 13.29113442297297485018934341455