| L(s) = 1 | − 1.73i·3-s + 4.37·5-s − 2.64i·7-s − 2.99·9-s − 3.58i·11-s − 7.58i·15-s − 2.55·17-s + 5.29i·19-s − 4.58·21-s + 0.417i·23-s + 14.1·25-s + 5.19i·27-s + 3.46i·31-s − 6.20·33-s − 11.5i·35-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + 1.95·5-s − 0.999i·7-s − 0.999·9-s − 1.08i·11-s − 1.95i·15-s − 0.618·17-s + 1.21i·19-s − 0.999·21-s + 0.0870i·23-s + 2.83·25-s + 0.999i·27-s + 0.622i·31-s − 1.08·33-s − 1.95i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38633 - 1.38633i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38633 - 1.38633i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 - 4.37T + 5T^{2} \) |
| 11 | \( 1 + 3.58iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 - 0.417iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 15.5iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.46713436428088052138502185223, −9.362893880337679950259420543162, −8.618577893810968009436798576914, −7.55603708101471493084271252287, −6.51081619210080034566811347873, −6.04901074155748437199257728809, −5.12205408001173349958463765746, −3.37973695673668757480552524471, −2.12453580861952486796448052438, −1.11857736872093081676544675762,
2.07502265437936819412105566243, 2.75565937701162677545875938018, 4.53871779734137336448874358864, 5.29075387614880487917229469963, 6.00675820408880898632170995411, 6.93388133771300652399431487372, 8.625548628709971614708251151045, 9.252203489606173559799210337401, 9.695857530439896998132700490773, 10.50095642638359302719529106217
