| L(s) = 1 | + (−1.83 − 0.784i)2-s + (2.76 + 2.88i)4-s − 6.85·5-s − 3.48i·7-s + (−2.82 − 7.48i)8-s + (12.6 + 5.37i)10-s − 1.50i·11-s + 0.937·13-s + (−2.73 + 6.40i)14-s + (−0.666 + 15.9i)16-s − 17.5·17-s − 4.35i·19-s + (−18.9 − 19.7i)20-s + (−1.18 + 2.76i)22-s − 9.27i·23-s + ⋯ |
| L(s) = 1 | + (−0.919 − 0.392i)2-s + (0.692 + 0.721i)4-s − 1.37·5-s − 0.497i·7-s + (−0.353 − 0.935i)8-s + (1.26 + 0.537i)10-s − 0.136i·11-s + 0.0721·13-s + (−0.195 + 0.457i)14-s + (−0.0416 + 0.999i)16-s − 1.03·17-s − 0.229i·19-s + (−0.949 − 0.989i)20-s + (−0.0536 + 0.125i)22-s − 0.403i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4991768145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4991768145\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.83 + 0.784i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 + 6.85T + 25T^{2} \) |
| 7 | \( 1 + 3.48iT - 49T^{2} \) |
| 11 | \( 1 + 1.50iT - 121T^{2} \) |
| 13 | \( 1 - 0.937T + 169T^{2} \) |
| 17 | \( 1 + 17.5T + 289T^{2} \) |
| 23 | \( 1 + 9.27iT - 529T^{2} \) |
| 29 | \( 1 + 4.00T + 841T^{2} \) |
| 31 | \( 1 + 21.0iT - 961T^{2} \) |
| 37 | \( 1 + 13.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 30.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 50.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 74.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 18.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 13.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 12.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 48.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 20.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 53.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 111. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 136.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 70.1T + 9.40e3T^{2} \) |
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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
−10.55441962558437707292834504315, −9.444457635393523105056732659456, −8.628802601531516757566032389167, −7.83920548727797377020956810481, −7.22772850761415529412579122397, −6.27742606627043117701039113851, −4.47778337021880250634323002003, −3.75138924264469409232766005300, −2.55257698118935121341093049887, −0.846550121257065103084175301757,
0.33930781253130400949208044174, 2.06227189854417163336839955159, 3.49106723339765048556847843169, 4.74010809285449007390382668444, 5.86683899754731624608671255533, 6.98176816861392813239587824053, 7.56463753187573880123121702539, 8.568916049565407040882883836671, 8.955614097339826229319077148111, 10.16953382290189970303313009590
