| L(s) = 1 | + (1.36 − 0.372i)2-s + (−3.30 + 0.828i)3-s + (1.72 − 1.01i)4-s + (0.400 + 0.189i)5-s + (−4.20 + 2.35i)6-s + (−1.08 − 3.59i)7-s + (1.97 − 2.02i)8-s + (7.59 − 4.06i)9-s + (0.617 + 0.109i)10-s + (4.48 − 0.665i)11-s + (−4.85 + 4.78i)12-s + (−0.364 + 1.01i)13-s + (−2.82 − 4.49i)14-s + (−1.48 − 0.294i)15-s + (1.93 − 3.49i)16-s + (1.44 − 0.288i)17-s + ⋯ |
| L(s) = 1 | + (0.964 − 0.263i)2-s + (−1.90 + 0.478i)3-s + (0.861 − 0.507i)4-s + (0.179 + 0.0847i)5-s + (−1.71 + 0.963i)6-s + (−0.411 − 1.35i)7-s + (0.697 − 0.716i)8-s + (2.53 − 1.35i)9-s + (0.195 + 0.0346i)10-s + (1.35 − 0.200i)11-s + (−1.40 + 1.38i)12-s + (−0.101 + 0.282i)13-s + (−0.754 − 1.20i)14-s + (−0.382 − 0.0760i)15-s + (0.484 − 0.874i)16-s + (0.351 − 0.0698i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15496 - 0.621276i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15496 - 0.621276i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 + 0.372i)T \) |
| good | 3 | \( 1 + (3.30 - 0.828i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-0.400 - 0.189i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (1.08 + 3.59i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (-4.48 + 0.665i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (0.364 - 1.01i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 0.288i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (1.25 + 1.13i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (5.33 - 0.525i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (-0.0286 - 0.0385i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-3.40 - 1.40i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.284 - 5.78i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-3.68 + 4.48i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (0.0410 - 0.163i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (-9.19 - 6.14i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (0.738 - 0.996i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (0.541 + 1.51i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (3.34 - 5.57i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-9.98 - 5.98i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (5.10 - 9.54i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (2.41 - 7.96i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-5.20 - 7.79i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.451 + 9.19i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (6.87 + 0.677i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-3.09 + 7.46i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.84915722901592871048138011014, −11.10837165544994834175956929629, −10.25351852512590371224867556679, −9.737846847519088821023564973409, −7.18506797349246811983069783564, −6.49149814054785513774305363236, −5.78597150848449666956588830923, −4.36491813542122154050821169361, −3.94809756537212694747554712738, −1.07243654435857766318179442559,
1.90900853448798164000248907806, 4.09161122676251711811004186714, 5.35898958335111885742486938493, 6.00285175671165029475402183200, 6.58453839378362485658353499632, 7.79013808841132602370718151045, 9.500152458451147220668486810531, 10.70687300537243577380107482423, 11.76444450462453410203935724567, 12.11316218900553054112540952941
