| 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
| \(L(1)\) |
\(\approx\) |
\(0.501784 + 0.484899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.501784 + 0.484899i\) |
| \(L(\frac{3}{2})\) |
|
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
−12.34311400670985931596754848265, −11.18578277389389697904213226603, −10.27126383217846842353228214566, −9.630019641807091201905040793708, −8.626710211926974859544209617170, −7.30790505257607122214211524993, −6.38820306523184060508178536038, −5.06966530297209793477294156218, −3.81089266153517650433016824059, −2.06634732829371572326563103330,
0.63785935165237415554850599101, 3.32130927185436354277099830122, 4.39450301541084139166757451515, 5.59890686609059142331509056542, 7.04560758329660796389877974175, 8.227622197184078193691998977458, 8.856596666339093222517658407507, 9.618535768383830151142293331805, 10.97321967139074796256243485580, 11.96879328246082458064912780422
