| L(s) = 1 | − i·2-s − 4-s + 2.84·5-s + (1.96 + 1.77i)7-s + i·8-s − 2.84i·10-s − 1.86i·11-s − 1.30i·13-s + (1.77 − 1.96i)14-s + 16-s + 1.41·17-s − 6.74i·19-s − 2.84·20-s − 1.86·22-s − 3.59i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.27·5-s + (0.741 + 0.671i)7-s + 0.353i·8-s − 0.900i·10-s − 0.562i·11-s − 0.362i·13-s + (0.474 − 0.524i)14-s + 0.250·16-s + 0.342·17-s − 1.54i·19-s − 0.636·20-s − 0.397·22-s − 0.749i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.414214982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.414214982\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.96 - 1.77i)T \) |
| 29 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.84T + 5T^{2} \) |
| 11 | \( 1 + 1.86iT - 11T^{2} \) |
| 13 | \( 1 + 1.30iT - 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 6.74iT - 19T^{2} \) |
| 23 | \( 1 + 3.59iT - 23T^{2} \) |
| 31 | \( 1 + 6.18iT - 31T^{2} \) |
| 37 | \( 1 + 0.842T + 37T^{2} \) |
| 41 | \( 1 + 4.97T + 41T^{2} \) |
| 43 | \( 1 + 7.57T + 43T^{2} \) |
| 47 | \( 1 + 0.221T + 47T^{2} \) |
| 53 | \( 1 - 7.02iT - 53T^{2} \) |
| 59 | \( 1 - 6.56T + 59T^{2} \) |
| 61 | \( 1 + 10.2iT - 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 + 8.62iT - 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 2.39T + 83T^{2} \) |
| 89 | \( 1 + 0.345T + 89T^{2} \) |
| 97 | \( 1 - 1.95iT - 97T^{2} \) |
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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
−8.511195781787169458236871243713, −7.86558619028278223989430148636, −6.66247940316390500267680648873, −5.96091905251039922745607295242, −5.20801197364526410185041993490, −4.71035683159772721047229125265, −3.41477970543229360086003061543, −2.48709592478878943218697475900, −1.94858812325426817973028891384, −0.72807868628100521528663255176,
1.35071511491014436312199376323, 1.94968000511058454514070192197, 3.43461099982844427936083654484, 4.28285160443229259538902129182, 5.30763090797405309641209824258, 5.55024957802413558128597446548, 6.67964400008951002304486678841, 7.06348073507934141035904425286, 8.055826853016596340365080723295, 8.521126953489614203587023328442
