| L(s) = 1 | − 2-s + 4-s + (−2.22 + 0.191i)5-s − 8-s + (2.22 − 0.191i)10-s + 2.39i·11-s + 5.67·13-s + 16-s + 2.07i·17-s + 5.91i·19-s + (−2.22 + 0.191i)20-s − 2.39i·22-s − 1.86·23-s + (4.92 − 0.852i)25-s − 5.67·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.996 + 0.0855i)5-s − 0.353·8-s + (0.704 − 0.0605i)10-s + 0.722i·11-s + 1.57·13-s + 0.250·16-s + 0.502i·17-s + 1.35i·19-s + (−0.498 + 0.0427i)20-s − 0.511i·22-s − 0.388·23-s + (0.985 − 0.170i)25-s − 1.11·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8932184592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8932184592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.22 - 0.191i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2.39iT - 11T^{2} \) |
| 13 | \( 1 - 5.67T + 13T^{2} \) |
| 17 | \( 1 - 2.07iT - 17T^{2} \) |
| 19 | \( 1 - 5.91iT - 19T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 + 4.88iT - 29T^{2} \) |
| 31 | \( 1 + 4.52iT - 31T^{2} \) |
| 37 | \( 1 - 2.96iT - 37T^{2} \) |
| 41 | \( 1 - 7.04T + 41T^{2} \) |
| 43 | \( 1 - 8.55iT - 43T^{2} \) |
| 47 | \( 1 - 5.57iT - 47T^{2} \) |
| 53 | \( 1 + 4.19T + 53T^{2} \) |
| 59 | \( 1 + 2.00T + 59T^{2} \) |
| 61 | \( 1 + 12.4iT - 61T^{2} \) |
| 67 | \( 1 + 7.62iT - 67T^{2} \) |
| 71 | \( 1 + 9.14iT - 71T^{2} \) |
| 73 | \( 1 - 1.08T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 13.6iT - 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.231876153461479424922016662175, −8.054909856377837128023642645424, −7.39891810694704740023988379023, −6.22988665825692129896686884834, −6.09804616104877650742321127955, −4.65005757040316813501616566498, −3.92467110399041396948000064250, −3.25744987770208081027165965100, −2.00312268829489838512688117314, −1.03485238732926539047806962317,
0.40425424988187079776725570228, 1.29673352146038408363259837430, 2.72830534587870051045343713258, 3.46838291825635856758639115872, 4.23776355819624942354950889460, 5.28096568628902679929277875956, 6.08471614380638267175182246861, 6.98470751662046016087850856242, 7.39332067526570739251533752394, 8.445896143652550752416471038159
