| L(s) = 1 | + (−1.78 − 0.907i)2-s + (2.35 + 3.23i)4-s + 3.42·5-s + 1.09i·7-s + (−1.26 − 7.89i)8-s + (−6.09 − 3.10i)10-s − 15.1i·11-s − 3.52·13-s + (0.995 − 1.95i)14-s + (−4.91 + 15.2i)16-s − 12.8·17-s − 4.35i·19-s + (8.05 + 11.0i)20-s + (−13.7 + 27.0i)22-s + 4.00i·23-s + ⋯ |
| L(s) = 1 | + (−0.891 − 0.453i)2-s + (0.588 + 0.808i)4-s + 0.684·5-s + 0.156i·7-s + (−0.157 − 0.987i)8-s + (−0.609 − 0.310i)10-s − 1.38i·11-s − 0.271·13-s + (0.0711 − 0.139i)14-s + (−0.307 + 0.951i)16-s − 0.757·17-s − 0.229i·19-s + (0.402 + 0.553i)20-s + (−0.626 + 1.23i)22-s + 0.174i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7284202122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7284202122\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.78 + 0.907i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 - 3.42T + 25T^{2} \) |
| 7 | \( 1 - 1.09iT - 49T^{2} \) |
| 11 | \( 1 + 15.1iT - 121T^{2} \) |
| 13 | \( 1 + 3.52T + 169T^{2} \) |
| 17 | \( 1 + 12.8T + 289T^{2} \) |
| 23 | \( 1 - 4.00iT - 529T^{2} \) |
| 29 | \( 1 + 42.9T + 841T^{2} \) |
| 31 | \( 1 + 36.9iT - 961T^{2} \) |
| 37 | \( 1 - 56.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.67T + 1.68e3T^{2} \) |
| 43 | \( 1 - 5.06iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 32.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 69.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 32.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 68.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 52.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 3.42iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 29.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 68.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 51.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 1.78T + 7.92e3T^{2} \) |
| 97 | \( 1 + 114.T + 9.40e3T^{2} \) |
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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
−9.676479286356043370219260829413, −9.309784085071271968333769866128, −8.342419648642025891368400046153, −7.55120991399338629398505191000, −6.37797651913942659635625911829, −5.67297127230611600058513822745, −4.07627542525536583899424651273, −2.88913779125321947618291284145, −1.86128058477410114623575473691, −0.33298011873088491785154388911,
1.54646172651985279515281540188, 2.48025018836314293759544100892, 4.36955885334510226208060372909, 5.42362564285405951431289083552, 6.37204638556673855235286515672, 7.20711535005888094309516437958, 7.924354298825120728421030272640, 9.148308984980245159838390312259, 9.588238192496934134224863437777, 10.37985794695257071957189984988
