L(s) = 1 | + 2.73i·3-s − 0.732i·5-s + 7-s − 4.46·9-s − 5.46i·11-s + 0.732i·13-s + 2·15-s + 0.535·17-s − 4.19i·19-s + 2.73i·21-s + 4.92·23-s + 4.46·25-s − 3.99i·27-s + 3.46i·29-s + 9.46·31-s + ⋯ |
L(s) = 1 | + 1.57i·3-s − 0.327i·5-s + 0.377·7-s − 1.48·9-s − 1.64i·11-s + 0.203i·13-s + 0.516·15-s + 0.129·17-s − 0.962i·19-s + 0.596i·21-s + 1.02·23-s + 0.892·25-s − 0.769i·27-s + 0.643i·29-s + 1.69·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.828539103\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.828539103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 2.73iT - 3T^{2} \) |
| 5 | \( 1 + 0.732iT - 5T^{2} \) |
| 11 | \( 1 + 5.46iT - 11T^{2} \) |
| 13 | \( 1 - 0.732iT - 13T^{2} \) |
| 17 | \( 1 - 0.535T + 17T^{2} \) |
| 19 | \( 1 + 4.19iT - 19T^{2} \) |
| 23 | \( 1 - 4.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 - 11.4iT - 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 9.46iT - 43T^{2} \) |
| 47 | \( 1 - 5.46T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4.19iT - 59T^{2} \) |
| 61 | \( 1 + 13.1iT - 61T^{2} \) |
| 67 | \( 1 - 1.07iT - 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 4.92T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 10.7iT - 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 - 7.46T + 97T^{2} \) |
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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
−9.183739273255392723639773872033, −8.813360184318377885480127935647, −8.219540337327791830991409424322, −6.89985728286161736770429246732, −5.94336769717905165787924854229, −4.96319786421136577335121990107, −4.67628803967934502731106463455, −3.45451295076912313358052218485, −2.88062210475824555636089253444, −0.899703280123874887803585800784,
1.03469505783266015755837440082, 2.02091744065489837613140076739, 2.81386405086547800903504012379, 4.25348205614624460703389397183, 5.23744675204222577953929710174, 6.26411950007931699428408471275, 6.90685299064400065435569443652, 7.56043625517431947587895587913, 8.053858181209434978460107615991, 9.036873106622873092506620995167