| L(s) = 1 | + (−0.508 + 1.31i)2-s + (0.0490 − 0.998i)3-s + (−1.48 − 1.34i)4-s + (2.27 − 3.80i)5-s + (1.29 + 0.572i)6-s + (2.92 − 3.56i)7-s + (2.52 − 1.27i)8-s + (−0.995 − 0.0980i)9-s + (3.85 + 4.94i)10-s + (0.473 + 0.169i)11-s + (−1.41 + 1.41i)12-s + (−0.399 + 1.59i)13-s + (3.21 + 5.67i)14-s + (−3.68 − 2.46i)15-s + (0.396 + 3.98i)16-s + (1.26 − 0.848i)17-s + ⋯ |
| L(s) = 1 | + (−0.359 + 0.933i)2-s + (0.0283 − 0.576i)3-s + (−0.741 − 0.671i)4-s + (1.01 − 1.70i)5-s + (0.527 + 0.233i)6-s + (1.10 − 1.34i)7-s + (0.892 − 0.450i)8-s + (−0.331 − 0.0326i)9-s + (1.22 + 1.56i)10-s + (0.142 + 0.0510i)11-s + (−0.408 + 0.408i)12-s + (−0.110 + 0.442i)13-s + (0.859 + 1.51i)14-s + (−0.951 − 0.635i)15-s + (0.0991 + 0.995i)16-s + (0.307 − 0.205i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33701 - 0.867634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33701 - 0.867634i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.508 - 1.31i)T \) |
| 3 | \( 1 + (-0.0490 + 0.998i)T \) |
| good | 5 | \( 1 + (-2.27 + 3.80i)T + (-2.35 - 4.40i)T^{2} \) |
| 7 | \( 1 + (-2.92 + 3.56i)T + (-1.36 - 6.86i)T^{2} \) |
| 11 | \( 1 + (-0.473 - 0.169i)T + (8.50 + 6.97i)T^{2} \) |
| 13 | \( 1 + (0.399 - 1.59i)T + (-11.4 - 6.12i)T^{2} \) |
| 17 | \( 1 + (-1.26 + 0.848i)T + (6.50 - 15.7i)T^{2} \) |
| 19 | \( 1 + (-0.0689 - 0.464i)T + (-18.1 + 5.51i)T^{2} \) |
| 23 | \( 1 + (-1.19 - 3.92i)T + (-19.1 + 12.7i)T^{2} \) |
| 29 | \( 1 + (0.824 - 1.74i)T + (-18.3 - 22.4i)T^{2} \) |
| 31 | \( 1 + (-1.47 - 3.56i)T + (-21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (2.41 - 1.78i)T + (10.7 - 35.4i)T^{2} \) |
| 41 | \( 1 + (2.08 - 3.89i)T + (-22.7 - 34.0i)T^{2} \) |
| 43 | \( 1 + (-3.83 + 0.188i)T + (42.7 - 4.21i)T^{2} \) |
| 47 | \( 1 + (2.53 - 12.7i)T + (-43.4 - 17.9i)T^{2} \) |
| 53 | \( 1 + (-3.30 - 6.99i)T + (-33.6 + 40.9i)T^{2} \) |
| 59 | \( 1 + (-2.32 - 9.30i)T + (-52.0 + 27.8i)T^{2} \) |
| 61 | \( 1 + (-10.5 + 9.53i)T + (5.97 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-6.13 - 6.76i)T + (-6.56 + 66.6i)T^{2} \) |
| 71 | \( 1 + (1.10 + 11.2i)T + (-69.6 + 13.8i)T^{2} \) |
| 73 | \( 1 + (7.03 + 8.57i)T + (-14.2 + 71.5i)T^{2} \) |
| 79 | \( 1 + (13.1 - 2.61i)T + (72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-9.85 - 7.31i)T + (24.0 + 79.4i)T^{2} \) |
| 89 | \( 1 + (0.892 - 2.94i)T + (-74.0 - 49.4i)T^{2} \) |
| 97 | \( 1 + (11.3 - 4.70i)T + (68.5 - 68.5i)T^{2} \) |
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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
−9.839460003819533219694140001140, −9.149414367969701446136304975521, −8.325002375876591313074577923143, −7.69779239388628227952463074496, −6.79168101367791077081453912253, −5.69336865754059073312322285911, −4.93890800079231698973746989940, −4.25305363145843276166371610487, −1.58294361594004505333434497950, −1.04742736960910936166048050284,
2.05966523376145518377872052292, 2.59128120967796180620168404743, 3.67578334037150141483808783782, 5.14551270621256639185799240194, 5.81992671641513982701251410175, 7.09983999473394762702610548766, 8.256712702706100230601183773067, 8.954815050192197866444699689186, 9.977334600569473321833841160604, 10.32508055232511611251834740301
