| L(s) = 1 | + (0.390 − 0.142i)2-s + (−1.39 + 1.17i)4-s + (0.384 + 2.18i)5-s + (−1.01 − 0.848i)7-s + (−0.795 + 1.37i)8-s + (0.460 + 0.797i)10-s + (−0.905 + 5.13i)11-s + (0.0169 + 0.00617i)13-s + (−0.515 − 0.187i)14-s + (0.519 − 2.94i)16-s + (1.56 + 2.71i)17-s + (−0.208 + 0.361i)19-s + (−3.10 − 2.60i)20-s + (0.376 + 2.13i)22-s + (−0.792 + 0.664i)23-s + ⋯ |
| L(s) = 1 | + (0.276 − 0.100i)2-s + (−0.699 + 0.587i)4-s + (0.172 + 0.975i)5-s + (−0.382 − 0.320i)7-s + (−0.281 + 0.486i)8-s + (0.145 + 0.252i)10-s + (−0.273 + 1.54i)11-s + (0.00470 + 0.00171i)13-s + (−0.137 − 0.0501i)14-s + (0.129 − 0.737i)16-s + (0.379 + 0.658i)17-s + (−0.0478 + 0.0829i)19-s + (−0.693 − 0.581i)20-s + (0.0802 + 0.454i)22-s + (−0.165 + 0.138i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0342 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.751130 + 0.777286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.751130 + 0.777286i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.390 + 0.142i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.384 - 2.18i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (1.01 + 0.848i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.905 - 5.13i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.0169 - 0.00617i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.56 - 2.71i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.208 - 0.361i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.792 - 0.664i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.33 + 2.67i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.85 + 2.39i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (2.21 + 3.83i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.45 - 1.25i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.44 - 8.18i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.43 + 4.56i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 1.30T + 53T^{2} \) |
| 59 | \( 1 + (0.642 + 3.64i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.29 - 4.44i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 3.77i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.04 - 5.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.273 + 0.473i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.459 - 0.167i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (4.33 - 1.57i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-1.68 + 2.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.72 + 9.79i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−12.52966611760209190054634134615, −11.53436946680918107663824265673, −10.20669517940179768214796467534, −9.772386084975495854040855474431, −8.338338980947163357605735285577, −7.37576868213439897856276577240, −6.39151019156706301110293167063, −4.88000354711851578735711260524, −3.79014165510334155319194330646, −2.53929463863001676388537566997,
0.852786274537696414959674656872, 3.22545817610469367938676014630, 4.75279974083994191648325169326, 5.52132025608320826266811281004, 6.48590175787726194640799805026, 8.303080061713132405104236979805, 8.893233266348335848148073448823, 9.803849203793051051682093935601, 10.83597869409827603309093474893, 12.11682413392137367394932372621
