| L(s) = 1 | + 0.529i·2-s + 1.71·4-s − 0.753·5-s + 1.96i·8-s − 0.399i·10-s + 5.38i·11-s − 1.46i·13-s + 2.39·16-s + 5.82·17-s − 3.54i·19-s − 1.29·20-s − 2.85·22-s + 3.26i·23-s − 4.43·25-s + 0.774·26-s + ⋯ |
| L(s) = 1 | + 0.374i·2-s + 0.859·4-s − 0.337·5-s + 0.696i·8-s − 0.126i·10-s + 1.62i·11-s − 0.405i·13-s + 0.599·16-s + 1.41·17-s − 0.813i·19-s − 0.289·20-s − 0.608·22-s + 0.681i·23-s − 0.886·25-s + 0.151·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.932960683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.932960683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.529iT - 2T^{2} \) |
| 5 | \( 1 + 0.753T + 5T^{2} \) |
| 11 | \( 1 - 5.38iT - 11T^{2} \) |
| 13 | \( 1 + 1.46iT - 13T^{2} \) |
| 17 | \( 1 - 5.82T + 17T^{2} \) |
| 19 | \( 1 + 3.54iT - 19T^{2} \) |
| 23 | \( 1 - 3.26iT - 23T^{2} \) |
| 29 | \( 1 + 0.0196iT - 29T^{2} \) |
| 31 | \( 1 - 6.49iT - 31T^{2} \) |
| 37 | \( 1 + 0.730T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.78T + 43T^{2} \) |
| 47 | \( 1 - 3.42T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 7.88T + 59T^{2} \) |
| 61 | \( 1 - 12.3iT - 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 13.6iT - 71T^{2} \) |
| 73 | \( 1 + 6.55iT - 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 1.63T + 89T^{2} \) |
| 97 | \( 1 - 16.8iT - 97T^{2} \) |
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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
−9.967567944189546036376280278477, −8.949514239108032866260749177238, −7.81664746815599380982941785643, −7.43441377583837831532458399016, −6.72414139282856097538651480816, −5.64641126152498228049421517279, −4.93226877739775674935420313378, −3.69395301820378060529251306137, −2.63169410815692306027728894195, −1.49149456147945054132125527663,
0.829625866630871717347947224257, 2.16341028609177326911033715533, 3.35352587095936950073214819750, 3.86307067196520221955048719996, 5.48280056676171378370732749986, 6.07315091066904520692353122243, 6.98353631977179107108941653226, 7.964575791547394745463911494045, 8.420561905826792937363135930422, 9.727249243539329208266472034712
