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
−8.036373277394840711304290861890, −7.39927599346309821139264241030, −6.78682087597299031707621658643, −6.05444906419648009923081083954, −5.18919877161774478975524232421, −4.29752388814128003882379858894, −3.65518204889205029934487265029, −2.51505388792759405527698783706, −1.76028861041294364612386140118, −0.092270500147766648861724307336,
1.06514341566642674159890387925, 2.50263979703275888401011401198, 3.39843716138584351607481233628, 3.80651490657805057591021458403, 5.16110440468915807620758249581, 5.78649755878343244722395957762, 6.45131740147855931575883484719, 7.12426478876648107163142813296, 8.086615692244498855291843033796, 8.777153374956352910575202989364