| L(s) = 1 | + 1.66i·3-s + 8.40i·5-s + 6.24·9-s + 5.21·11-s − 4.88i·13-s − 13.9·15-s − 7.72i·17-s + 35.3i·19-s + 26.6·23-s − 45.5·25-s + 25.3i·27-s + 45.1·29-s + 40.4i·31-s + 8.66i·33-s + 7.95·37-s + ⋯ |
| L(s) = 1 | + 0.553i·3-s + 1.68i·5-s + 0.693·9-s + 0.474·11-s − 0.375i·13-s − 0.930·15-s − 0.454i·17-s + 1.85i·19-s + 1.15·23-s − 1.82·25-s + 0.937i·27-s + 1.55·29-s + 1.30i·31-s + 0.262i·33-s + 0.215·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.192802755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192802755\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.66iT - 9T^{2} \) |
| 5 | \( 1 - 8.40iT - 25T^{2} \) |
| 11 | \( 1 - 5.21T + 121T^{2} \) |
| 13 | \( 1 + 4.88iT - 169T^{2} \) |
| 17 | \( 1 + 7.72iT - 289T^{2} \) |
| 19 | \( 1 - 35.3iT - 361T^{2} \) |
| 23 | \( 1 - 26.6T + 529T^{2} \) |
| 29 | \( 1 - 45.1T + 841T^{2} \) |
| 31 | \( 1 - 40.4iT - 961T^{2} \) |
| 37 | \( 1 - 7.95T + 1.36e3T^{2} \) |
| 41 | \( 1 - 26.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 0.403T + 1.84e3T^{2} \) |
| 47 | \( 1 + 32.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 81.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 8.80iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 29.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 87.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 27.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 87.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 121.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 46.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 60.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 66.4iT - 9.40e3T^{2} \) |
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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
−9.867427541814012165861442104239, −8.843265668108467497533273892211, −7.82070022028745322772518471523, −7.01813229334919236051614655125, −6.51230349692773747615833640328, −5.49702195569181929250729151994, −4.38967182858793012877125781203, −3.47068628353520088676254961179, −2.84113969657436648970046148866, −1.42246016896604995340753448144,
0.67905894684598525734655108864, 1.33374296722161242431860654667, 2.55771743123110396495289064670, 4.22131434657207153580019000964, 4.59623343238581389461861419426, 5.58378438289946956694313074765, 6.63186888424818902845913654372, 7.29606732933512985480363463960, 8.277221420995205663048967684862, 8.953807745562867847549350389486
