L(s) = 1 | + (−0.347 + 1.96i)4-s + (−0.694 − 3.93i)7-s + (−5.36 − 4.49i)13-s + (−3.75 − 1.36i)16-s + (0.5 + 0.866i)19-s + (−3.83 + 3.21i)25-s + 7.99·28-s + (1.91 − 10.8i)31-s + (5 − 8.66i)37-s + (−4.69 − 1.71i)43-s + (−8.45 + 3.07i)49-s + (10.7 − 8.99i)52-s + (−0.173 − 0.984i)61-s + (4 − 6.92i)64-s + (3.83 + 3.21i)67-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)4-s + (−0.262 − 1.48i)7-s + (−1.48 − 1.24i)13-s + (−0.939 − 0.342i)16-s + (0.114 + 0.198i)19-s + (−0.766 + 0.642i)25-s + 1.51·28-s + (0.343 − 1.94i)31-s + (0.821 − 1.42i)37-s + (−0.716 − 0.260i)43-s + (−1.20 + 0.439i)49-s + (1.48 − 1.24i)52-s + (−0.0222 − 0.126i)61-s + (0.5 − 0.866i)64-s + (0.467 + 0.392i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.446000 - 0.599083i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.446000 - 0.599083i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.694 + 3.93i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (5.36 + 4.49i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.91 + 10.8i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (4.69 + 1.71i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.173 + 0.984i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.83 - 3.21i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.95 - 8.35i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.69 + 1.71i)T + (74.3 + 62.3i)T^{2} \) |
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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
−9.996477807193794753705179535360, −9.504665808894277382728136680681, −8.074170725257147030203583753843, −7.60037640200099998854790644840, −6.98567091502667497329450257378, −5.58629501134884344251630837262, −4.39226810190590639148762898278, −3.65892967745008029953096080381, −2.53487223280189715835326118140, −0.36754890654905086576193349311,
1.81409665349745421782827441751, 2.78895331319813385966778326699, 4.57828893963226823238894653893, 5.20201244562254585610395849962, 6.22869250899549937937463422121, 6.88696430194577783622817440697, 8.275484792753622714708764787517, 9.180583112405194193077904577244, 9.646660494873775946032257249295, 10.45429476993624832380300033315