| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·8-s − 9-s − 6·11-s + i·12-s − 2i·13-s + 16-s + 4i·17-s + i·18-s + 6i·22-s + 2i·23-s + 24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s − 1.80·11-s + 0.288i·12-s − 0.554i·13-s + 0.250·16-s + 0.970i·17-s + 0.235i·18-s + 1.27i·22-s + 0.417i·23-s + 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.280022608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280022608\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
| 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
−8.215521347488716727065147828818, −7.69453177427700279241204865678, −6.76882594093222381022499700857, −5.85880405287366752514654024319, −5.23386823113509159919124277167, −4.45980559807507340923530493224, −3.29084978940185802728991071371, −2.71051603887472599889862779391, −1.82243101039806623275622768745, −0.66349852205173712384211568619,
0.56280651768258827550041248441, 2.35446447170718947117765349837, 3.05164022062927671555517840733, 4.19971653465469856887402085289, 4.86097226009617162398738549509, 5.40765709812041091204603713941, 6.19776005159768904999767771728, 7.09761354352832094788790391309, 7.66969011805510670821495513065, 8.428323139521323225342691785709
