| L(s) = 1 | − 5.19i·3-s − 16.8i·7-s − 27·9-s + 152. i·11-s + 37·13-s − 27.8·17-s + 426. i·19-s − 87.4·21-s − 381. i·23-s + 140. i·27-s − 681.·29-s − 203. i·31-s + 793.·33-s − 662.·37-s − 192. i·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.343i·7-s − 0.333·9-s + 1.26i·11-s + 0.218·13-s − 0.0963·17-s + 1.18i·19-s − 0.198·21-s − 0.721i·23-s + 0.192i·27-s − 0.810·29-s − 0.211i·31-s + 0.728·33-s − 0.483·37-s − 0.126i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8061237852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8061237852\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 16.8iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 152. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 37T + 2.85e4T^{2} \) |
| 17 | \( 1 + 27.8T + 8.35e4T^{2} \) |
| 19 | \( 1 - 426. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 381. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 681.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 203. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 662.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.31e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 853. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.21e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 692.T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.91e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.03e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.97e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.08e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.21e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.83e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 69.4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 452.T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.31e4T + 8.85e7T^{2} \) |
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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
−8.805816081126692104402721154815, −7.86873221173357356183029399298, −7.27440578393487997326153310272, −6.46204846821071375158260302146, −5.56083872756801031921137274749, −4.49986654459200956642224409803, −3.61266378048118148659477016322, −2.28919385468314251369209304294, −1.48534080900567899394600470862, −0.17474583590825578507878445826,
1.05873614091597372603097019693, 2.55032702884416149881813979178, 3.39094416119667468429394810215, 4.34939409165478548894774986316, 5.41402800063537184846255207057, 5.98540705650319543096118648063, 7.06153837282169975748197965982, 8.047067597846832507802677350360, 8.939459154435646099622374104641, 9.298647496538432689264638318025
