| L(s) = 1 | + (−1.69 − 0.775i)2-s + (−0.953 + 1.44i)3-s + (0.975 + 1.12i)4-s + (3.49 − 2.24i)5-s + (2.74 − 1.71i)6-s + (2.15 + 0.309i)7-s + (0.268 + 0.915i)8-s + (−1.18 − 2.75i)9-s + (−7.68 + 1.10i)10-s + (−0.121 − 0.266i)11-s + (−2.55 + 0.336i)12-s + (0.183 + 1.27i)13-s + (−3.41 − 2.19i)14-s + (−0.0866 + 7.19i)15-s + (0.677 − 4.71i)16-s + (−0.579 + 0.668i)17-s + ⋯ |
| L(s) = 1 | + (−1.20 − 0.548i)2-s + (−0.550 + 0.834i)3-s + (0.487 + 0.562i)4-s + (1.56 − 1.00i)5-s + (1.11 − 0.700i)6-s + (0.813 + 0.116i)7-s + (0.0950 + 0.323i)8-s + (−0.393 − 0.919i)9-s + (−2.42 + 0.349i)10-s + (−0.0367 − 0.0804i)11-s + (−0.738 + 0.0970i)12-s + (0.0507 + 0.353i)13-s + (−0.912 − 0.586i)14-s + (−0.0223 + 1.85i)15-s + (0.169 − 1.17i)16-s + (−0.140 + 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.554381 - 0.122713i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.554381 - 0.122713i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.953 - 1.44i)T \) |
| 23 | \( 1 + (3.56 - 3.20i)T \) |
| good | 2 | \( 1 + (1.69 + 0.775i)T + (1.30 + 1.51i)T^{2} \) |
| 5 | \( 1 + (-3.49 + 2.24i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (-2.15 - 0.309i)T + (6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (0.121 + 0.266i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.183 - 1.27i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.579 - 0.668i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.11 - 0.968i)T + (2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (4.42 + 3.83i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (3.42 - 1.00i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (3.43 - 5.34i)T + (-15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-3.23 - 5.03i)T + (-17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (1.29 - 4.41i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 5.56iT - 47T^{2} \) |
| 53 | \( 1 + (0.572 - 3.97i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-2.75 + 0.395i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (0.616 + 2.09i)T + (-51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (10.9 + 4.98i)T + (43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (3.93 + 1.79i)T + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-3.76 - 4.35i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-13.1 + 1.89i)T + (75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (3.35 + 2.15i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-7.51 - 2.20i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (7.03 + 10.9i)T + (-40.2 + 88.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.70286141191645074681019989946, −13.57024895072143063948228381772, −12.05358769418499119858417684296, −10.98324576003994428836553638754, −9.934423195057880758211092664894, −9.299528000815637509820119325843, −8.336630383963110962990390089401, −5.89271713405101957699210324967, −4.83983820841043461475396406744, −1.72259311679737044884597950074,
1.93789656210423167918688403361, 5.62264477331245118971107074044, 6.70888338061653932269709090491, 7.62442200854118475451472738739, 9.015300915969689673192465192783, 10.34928073750145940082628273152, 10.98311037825667757878918012067, 12.77471719722815773807695602617, 13.83017920212526983448604331216, 14.76631372276175849542066737926
