| L(s) = 1 | − 2.64·7-s − 6.57i·11-s − 1.91i·23-s − 5·25-s + 8.89i·29-s − 10.5·37-s + 5.29·43-s + 7.00·49-s + 0.412i·53-s + 4·67-s + 15.0i·71-s + 17.3i·77-s + 8·79-s + 10.4i·107-s − 10.5·109-s + ⋯ |
| L(s) = 1 | − 0.999·7-s − 1.98i·11-s − 0.399i·23-s − 25-s + 1.65i·29-s − 1.73·37-s + 0.806·43-s + 49-s + 0.0566i·53-s + 0.488·67-s + 1.78i·71-s + 1.98i·77-s + 0.900·79-s + 1.00i·107-s − 1.01·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2985389924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2985389924\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 6.57iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 1.91iT - 23T^{2} \) |
| 29 | \( 1 - 8.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 0.412iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 15.0iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 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
−8.777153374956352910575202989364, −8.086615692244498855291843033796, −7.12426478876648107163142813296, −6.45131740147855931575883484719, −5.78649755878343244722395957762, −5.16110440468915807620758249581, −3.80651490657805057591021458403, −3.39843716138584351607481233628, −2.50263979703275888401011401198, −1.06514341566642674159890387925,
0.092270500147766648861724307336, 1.76028861041294364612386140118, 2.51505388792759405527698783706, 3.65518204889205029934487265029, 4.29752388814128003882379858894, 5.18919877161774478975524232421, 6.05444906419648009923081083954, 6.78682087597299031707621658643, 7.39927599346309821139264241030, 8.036373277394840711304290861890
