| L(s) = 1 | + 0.874i·3-s − 3.70·5-s + (−1.41 + 2.23i)7-s + 2.23·9-s − 3.23·11-s − 0.874·13-s − 3.23i·15-s + 4.57i·17-s − 1.95i·19-s + (−1.95 − 1.23i)21-s + 1.23i·23-s + 8.70·25-s + 4.57i·27-s + 2i·29-s − 10.2·31-s + ⋯ |
| L(s) = 1 | + 0.504i·3-s − 1.65·5-s + (−0.534 + 0.845i)7-s + 0.745·9-s − 0.975·11-s − 0.242·13-s − 0.835i·15-s + 1.10i·17-s − 0.448i·19-s + (−0.426 − 0.269i)21-s + 0.257i·23-s + 1.74·25-s + 0.880i·27-s + 0.371i·29-s − 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3198084467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3198084467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.41 - 2.23i)T \) |
| good | 3 | \( 1 - 0.874iT - 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 + 0.874T + 13T^{2} \) |
| 17 | \( 1 - 4.57iT - 17T^{2} \) |
| 19 | \( 1 + 1.95iT - 19T^{2} \) |
| 23 | \( 1 - 1.23iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 10.9iT - 37T^{2} \) |
| 41 | \( 1 + 3.90iT - 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 + 0.472iT - 53T^{2} \) |
| 59 | \( 1 + 11.1iT - 59T^{2} \) |
| 61 | \( 1 - 8.94T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 + 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 12.4iT - 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 9.82iT - 89T^{2} \) |
| 97 | \( 1 + 4.57iT - 97T^{2} \) |
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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
−9.030590311476525074893301349267, −8.371038236044634670118082855578, −7.48714862269339594314696910286, −7.01613930351058526005656702536, −5.70006648283570071394092509012, −4.95652685714819557759535732950, −3.90965075793879032396067784441, −3.47330818882741549023142934843, −2.16750271224823091257697284772, −0.15053470445527033493281457802,
0.956338335179852325575628291956, 2.65502475108992766539414905594, 3.66805248865687938681057844796, 4.34082049636174612501230860460, 5.24322757042736269571002733583, 6.65558020336422392750135371075, 7.24949679087065026063272197811, 7.68903049959574704907498123183, 8.340168101917413827755334927867, 9.525435978538232081021593401855
