| L(s) = 1 | + (−1.37 − 1.58i)2-s + (0.841 + 0.540i)3-s + (−0.343 + 2.39i)4-s + (1.43 − 3.13i)5-s + (−0.299 − 2.08i)6-s + (−1.51 − 0.445i)7-s + (0.736 − 0.473i)8-s + (0.415 + 0.909i)9-s + (−6.95 + 2.04i)10-s + (−0.903 + 1.04i)11-s + (−1.58 + 1.82i)12-s + (4.36 − 1.28i)13-s + (1.38 + 3.02i)14-s + (2.90 − 1.86i)15-s + (2.87 + 0.843i)16-s + (0.575 + 4.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.973 − 1.12i)2-s + (0.485 + 0.312i)3-s + (−0.171 + 1.19i)4-s + (0.640 − 1.40i)5-s + (−0.122 − 0.849i)6-s + (−0.573 − 0.168i)7-s + (0.260 − 0.167i)8-s + (0.138 + 0.303i)9-s + (−2.19 + 0.645i)10-s + (−0.272 + 0.314i)11-s + (−0.456 + 0.527i)12-s + (1.21 − 0.355i)13-s + (0.369 + 0.808i)14-s + (0.749 − 0.481i)15-s + (0.717 + 0.210i)16-s + (0.139 + 0.971i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0590 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0590 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.472125 - 0.500855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.472125 - 0.500855i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (3.84 + 2.87i)T \) |
| good | 2 | \( 1 + (1.37 + 1.58i)T + (-0.284 + 1.97i)T^{2} \) |
| 5 | \( 1 + (-1.43 + 3.13i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (1.51 + 0.445i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (0.903 - 1.04i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-4.36 + 1.28i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.575 - 4.00i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.869 - 6.04i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.43 - 9.98i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.956 + 0.614i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (3.43 + 7.51i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.03 + 4.45i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (2.41 + 1.55i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 4.66T + 47T^{2} \) |
| 53 | \( 1 + (6.27 + 1.84i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (0.908 - 0.266i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.641 + 0.412i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.82 - 2.11i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-3.71 - 4.28i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.129 - 0.900i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (9.35 - 2.74i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (2.01 + 4.41i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (3.08 + 1.98i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (0.708 - 1.55i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−14.26286704077122942709497856865, −12.84706927993454871270891833444, −12.46334650803916086623998747902, −10.64559007057663691006538798143, −9.967835089126149693486245391531, −8.859964269833065074308360791773, −8.268129788896047433404481049000, −5.74794865217811732594932805909, −3.73274870472234246596547930638, −1.65095493593113974458117140054,
2.96610049603749321884033439911, 6.12841387472057521702429235355, 6.73055672922555526910756018572, 7.925485990286303363714500782070, 9.209018505581879445062632818593, 10.05081188565535653546562069105, 11.43115649537210574986848539576, 13.43160995198206568103154386881, 14.07251208185288358109148330271, 15.37666876923691203833976756770
