| L(s) = 1 | + 1.41·2-s + 2.00·4-s − 4.24·5-s + 4i·7-s + 2.82·8-s − 6·10-s − 16.9·11-s + 5.65i·14-s + 4.00·16-s − 12.7i·17-s − 16i·19-s − 8.48·20-s − 24·22-s − 16.9i·23-s − 7.00·25-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.848·5-s + 0.571i·7-s + 0.353·8-s − 0.600·10-s − 1.54·11-s + 0.404i·14-s + 0.250·16-s − 0.748i·17-s − 0.842i·19-s − 0.424·20-s − 1.09·22-s − 0.737i·23-s − 0.280·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.137707868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137707868\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4.24T + 25T^{2} \) |
| 7 | \( 1 - 4iT - 49T^{2} \) |
| 11 | \( 1 + 16.9T + 121T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 + 16iT - 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 + 4.24iT - 841T^{2} \) |
| 31 | \( 1 - 44iT - 961T^{2} \) |
| 37 | \( 1 - 34iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 - 84.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 38.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 50T + 3.72e3T^{2} \) |
| 67 | \( 1 - 8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 16iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 76T + 6.24e3T^{2} \) |
| 83 | \( 1 - 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 12.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 176iT - 9.40e3T^{2} \) |
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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
−8.478760014607011104604431563708, −7.66727822660769770713295439399, −7.19465836621541399162300331903, −6.21096660633880117352519891817, −5.31380167132124344232555990027, −4.81096891969065607648741076824, −3.92635522649124511884964300000, −2.82771598589432305667952975407, −2.42322944606132668031744431966, −0.67647684726417969309038710230,
0.56375073860039615258006326154, 2.02201205579488614564001901635, 2.98413124066148647876116457851, 4.03002751155738656274900594519, 4.27134176033354957798204365956, 5.63287017159554483869970204737, 5.84692744474499414084674711373, 7.26197480326346149713298597586, 7.62650277005579897160359743379, 8.129513606302001056247406798484
