L(s) = 1 | + 2.82i·7-s + 5.65·11-s − 2i·13-s + 2i·17-s − 2.82i·23-s + 6·29-s + 5.65·31-s + 10i·37-s − 2·41-s − 8.48i·43-s − 2.82i·47-s − 1.00·49-s − 6i·53-s + 11.3·59-s − 2·61-s + ⋯ |
L(s) = 1 | + 1.06i·7-s + 1.70·11-s − 0.554i·13-s + 0.485i·17-s − 0.589i·23-s + 1.11·29-s + 1.01·31-s + 1.64i·37-s − 0.312·41-s − 1.29i·43-s − 0.412i·47-s − 0.142·49-s − 0.824i·53-s + 1.47·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.422324459\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.422324459\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2.82iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.294882035856602065400494184806, −7.12887697395578414376017763315, −6.46300188612343977543817865246, −6.04295934059458597521575693414, −5.15305494852376585102118142553, −4.42686598211880121812867009323, −3.57835621788388802753566999296, −2.79012952573848362269297956972, −1.87085214321009233565149092071, −0.879592843478337469356589750622,
0.812763391941497603436675494074, 1.49790607187681915886781130889, 2.69922534532862369397671722693, 3.70939780375753892015849306796, 4.20340810293441425257150978015, 4.83531634803333086873164805530, 5.96933656988545351079685409537, 6.55632892010217361749691370550, 7.13928390638840055020778815645, 7.70860452663239109555809312237