| L(s) = 1 | − 2.30i·3-s + 0.697i·7-s − 2.30·9-s + 11-s + 5i·13-s − 6.90i·17-s − 19-s + 1.60·21-s + 7.30i·23-s − 1.60i·27-s − 0.908·29-s − 10.2·31-s − 2.30i·33-s − 2.39i·37-s + 11.5·39-s + ⋯ |
| L(s) = 1 | − 1.32i·3-s + 0.263i·7-s − 0.767·9-s + 0.301·11-s + 1.38i·13-s − 1.67i·17-s − 0.229·19-s + 0.350·21-s + 1.52i·23-s − 0.308i·27-s − 0.168·29-s − 1.83·31-s − 0.400i·33-s − 0.393i·37-s + 1.84·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5448729080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5448729080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.30iT - 3T^{2} \) |
| 7 | \( 1 - 0.697iT - 7T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + 6.90iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 7.30iT - 23T^{2} \) |
| 29 | \( 1 + 0.908T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 2.39iT - 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 + 7.21iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 1.30iT - 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 2.60T + 71T^{2} \) |
| 73 | \( 1 + 7.90iT - 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 3.51iT - 83T^{2} \) |
| 89 | \( 1 + 1.69T + 89T^{2} \) |
| 97 | \( 1 + 15.3iT - 97T^{2} \) |
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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
−7.54771274267412604289334493151, −7.30119083990446843910728237487, −6.70827367924123355787588641631, −5.86180425310284841242924383360, −5.13453185508180892564053720610, −4.11717817004201581867908733121, −3.14394703366650379974078012920, −2.03464063972095173927573005295, −1.54911034786201867711883050409, −0.14525844383097415859734621194,
1.44422867953277196557160256412, 2.79736317887975206199752263782, 3.63509263252700718754121583203, 4.19207400774589290494702632423, 4.95636910060113215603955484263, 5.75929933302243275591941041872, 6.39556080759059731248627222109, 7.45036528656787311334627566807, 8.231052490416285258746349362156, 8.802172247308509300639920198705
