| L(s) = 1 | − 1.39·5-s − 4.57·7-s − 3.16i·11-s + 0.678i·13-s + 2.04·17-s − 3.83·19-s + 1.63·23-s − 3.05·25-s + 2.49i·29-s − 4.99·31-s + 6.38·35-s − 10.1i·37-s − 12.1·41-s − 4.71i·43-s + 3.25i·47-s + ⋯ |
| L(s) = 1 | − 0.623·5-s − 1.72·7-s − 0.954i·11-s + 0.188i·13-s + 0.496·17-s − 0.880·19-s + 0.341·23-s − 0.610·25-s + 0.462i·29-s − 0.897·31-s + 1.07·35-s − 1.66i·37-s − 1.90·41-s − 0.718i·43-s + 0.474i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5497860492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5497860492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + (-1.96 - 12.7i)T \) |
| good | 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 + 3.16iT - 11T^{2} \) |
| 13 | \( 1 - 0.678iT - 13T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 - 2.49iT - 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 + 10.1iT - 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 4.71iT - 43T^{2} \) |
| 47 | \( 1 - 3.25iT - 47T^{2} \) |
| 53 | \( 1 + 7.08T + 53T^{2} \) |
| 59 | \( 1 - 5.99T + 59T^{2} \) |
| 61 | \( 1 - 9.50T + 61T^{2} \) |
| 67 | \( 1 + 9.54iT - 67T^{2} \) |
| 71 | \( 1 + 9.35T + 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 17.2iT - 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 15.3iT - 89T^{2} \) |
| 97 | \( 1 - 7.44T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.332454123664649417702035911604, −7.20889987609047392710414249081, −6.91461733771650293532941252851, −5.95879266363325998122241005920, −5.57035203304054364551158148891, −4.32947876123866427065157498864, −3.51045899960325218459238883469, −3.24369911661550562050419633030, −2.05809864386048132597983061806, −0.54375443681727721513741202358,
0.25516047578481339351130588486, 1.75965753270208472938840098343, 2.88008235646038832511593471909, 3.50299038552410243927564668954, 4.21345389814705475001091534097, 5.08110157328541708790110337988, 6.02033119654278469785451956962, 6.65320332868785109093048532685, 7.17421617154991516519847035080, 7.946909023880845495813617736558
