| L(s) = 1 | − i·2-s + 3-s − 4-s + 0.339i·5-s − i·6-s + i·8-s + 9-s + 0.339·10-s + 0.660i·11-s − 12-s + (0.660 + 3.54i)13-s + 0.339i·15-s + 16-s − 6.54·17-s − i·18-s − 6.08i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.151i·5-s − 0.408i·6-s + 0.353i·8-s + 0.333·9-s + 0.107·10-s + 0.199i·11-s − 0.288·12-s + (0.183 + 0.983i)13-s + 0.0877i·15-s + 0.250·16-s − 1.58·17-s − 0.235i·18-s − 1.39i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8995331392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8995331392\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (-0.660 - 3.54i)T \) |
| good | 5 | \( 1 - 0.339iT - 5T^{2} \) |
| 11 | \( 1 - 0.660iT - 11T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 19 | \( 1 + 6.08iT - 19T^{2} \) |
| 23 | \( 1 + 7.42T + 23T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 3.54iT - 37T^{2} \) |
| 41 | \( 1 + 0.864iT - 41T^{2} \) |
| 43 | \( 1 + 5.08T + 43T^{2} \) |
| 47 | \( 1 + 9.44iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 3.90iT - 59T^{2} \) |
| 61 | \( 1 - 1.86T + 61T^{2} \) |
| 67 | \( 1 + 7.44iT - 67T^{2} \) |
| 71 | \( 1 - 0.884iT - 71T^{2} \) |
| 73 | \( 1 + 10.0iT - 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 16.2iT - 83T^{2} \) |
| 89 | \( 1 - 4.45iT - 89T^{2} \) |
| 97 | \( 1 + 16.1iT - 97T^{2} \) |
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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
−8.527912462437369555409654637339, −7.35952246544222616706359455279, −6.83245142865421411649003555141, −5.94533972855645330502335817210, −4.71514410312913586383785940794, −4.28199857800729156904164054905, −3.39576659042695160522484741526, −2.32358284802627463460168596012, −1.88504143856976079789902527145, −0.23121871138840340927849525644,
1.35845661465048565093303830234, 2.53490003680896979205702843015, 3.57110411770750429731181970302, 4.25331121433082607665052151700, 5.16729579029632355572695816750, 5.97005145785196015619808754546, 6.62877274296482041361296117987, 7.46409491141511617108424016930, 8.328683566484221890471769305390, 8.428460773574752649223750785881
