| L(s) = 1 | − 1.36·5-s − 1.24i·7-s + 5.79i·11-s + 16.3·13-s + 5.01·17-s − 26.1i·19-s + 25.1i·23-s − 23.1·25-s − 32.7·29-s + 1.01i·31-s + 1.69i·35-s + 14.9·37-s + 72.5·41-s − 33.4i·43-s + 66.5i·47-s + ⋯ |
| L(s) = 1 | − 0.272·5-s − 0.177i·7-s + 0.527i·11-s + 1.26·13-s + 0.294·17-s − 1.37i·19-s + 1.09i·23-s − 0.925·25-s − 1.13·29-s + 0.0328i·31-s + 0.0484i·35-s + 0.405·37-s + 1.76·41-s − 0.778i·43-s + 1.41i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.875673381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875673381\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.36T + 25T^{2} \) |
| 7 | \( 1 + 1.24iT - 49T^{2} \) |
| 11 | \( 1 - 5.79iT - 121T^{2} \) |
| 13 | \( 1 - 16.3T + 169T^{2} \) |
| 17 | \( 1 - 5.01T + 289T^{2} \) |
| 19 | \( 1 + 26.1iT - 361T^{2} \) |
| 23 | \( 1 - 25.1iT - 529T^{2} \) |
| 29 | \( 1 + 32.7T + 841T^{2} \) |
| 31 | \( 1 - 1.01iT - 961T^{2} \) |
| 37 | \( 1 - 14.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 72.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 33.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 54.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 20.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 111.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 80.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 30.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 80.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 113. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 21.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 160.T + 9.40e3T^{2} \) |
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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
−9.439441153328763390807525652845, −8.950428484368054658405052938432, −7.79458816146573854366651745483, −7.29636987965127851744640035833, −6.21053855379094370175661528801, −5.40425837207538797275693623409, −4.24950571247329485898469479740, −3.52097573420510630309383544807, −2.19320733040621285940007972935, −0.844540956673570599805990099870,
0.839286500957290092884434517454, 2.20169463184827016092484201040, 3.56725998553573016286078983482, 4.13396075816498622508237174458, 5.63814545760179999911842420052, 6.01948525711208024856791782243, 7.17720847507089163469398469179, 8.142484082134752177430138879517, 8.602056125159411752284459172968, 9.622396972042593862974765094895
