| L(s) = 1 | − 12.6i·3-s − 23.1i·5-s + (6.45 − 48.5i)7-s − 79.9·9-s − 191.·11-s − 48.5i·13-s − 294.·15-s + 181. i·17-s + 599. i·19-s + (−616. − 81.9i)21-s − 469.·23-s + 86.8·25-s − 13.0i·27-s + 338.·29-s − 267. i·31-s + ⋯ |
| L(s) = 1 | − 1.40i·3-s − 0.927i·5-s + (0.131 − 0.991i)7-s − 0.987·9-s − 1.58·11-s − 0.287i·13-s − 1.30·15-s + 0.629i·17-s + 1.66i·19-s + (−1.39 − 0.185i)21-s − 0.887·23-s + 0.138·25-s − 0.0179i·27-s + 0.402·29-s − 0.278i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4892401324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4892401324\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-6.45 + 48.5i)T \) |
| good | 3 | \( 1 + 12.6iT - 81T^{2} \) |
| 5 | \( 1 + 23.1iT - 625T^{2} \) |
| 11 | \( 1 + 191.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 48.5iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 181. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 599. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 469.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 338.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 267. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 668.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.32e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.94e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.93e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.46e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.73e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 246. iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 1.07e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.27e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 7.10e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 7.01e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.44e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 2.13e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.89e3iT - 8.85e7T^{2} \) |
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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
−9.994683180321941082830523240884, −8.249168631868654206687142540617, −8.103373872555252091450000199511, −7.18491609757673151253641958585, −6.05322883155413170971135887510, −5.11526074909850544364981277598, −3.80063155314878208937102648217, −2.17283793279220840085404143029, −1.16083330769771848997412355039, −0.13716999809232537602215420346,
2.52611819754367760592639566194, 3.08404114633171324939053081703, 4.62639335634641481380409522840, 5.21897683441617812187078184472, 6.39171168200024475366341494446, 7.59698633578780489946841116054, 8.696837037769268136339743142264, 9.529161136646600611123259716383, 10.30278013871484237413521856663, 10.99918530417194037590401512309
