| L(s) = 1 | + (−1.28 − 0.581i)2-s + (1.32 + 1.50i)4-s + (−0.832 − 2.70i)8-s − 0.913·11-s + (−0.5 + 3.96i)16-s + (1.17 + 0.531i)22-s − 9.39·23-s + 5·25-s − 8.89i·29-s + (2.95 − 4.82i)32-s − 10.5·37-s + 12i·43-s + (−1.20 − 1.36i)44-s + (12.1 + 5.46i)46-s + (−6.44 − 2.90i)50-s + ⋯ |
| L(s) = 1 | + (−0.911 − 0.411i)2-s + (0.661 + 0.750i)4-s + (−0.294 − 0.955i)8-s − 0.275·11-s + (−0.125 + 0.992i)16-s + (0.250 + 0.113i)22-s − 1.95·23-s + 25-s − 1.65i·29-s + (0.522 − 0.852i)32-s − 1.73·37-s + 1.82i·43-s + (−0.182 − 0.206i)44-s + (1.78 + 0.806i)46-s + (−0.911 − 0.411i)50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2399220165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2399220165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 + 0.581i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 0.913T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 9.39T + 23T^{2} \) |
| 29 | \( 1 + 8.89iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 0.412iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 7.57T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.968395324861851664399459137709, −8.117705452656967152059034211365, −7.66710721913306401771816600856, −6.61423751991571727520757406099, −5.93869798228090701208701978884, −4.63027926540045550014471656302, −3.65144859355296760302242581104, −2.63173646203382058077457704177, −1.63726291331363011581626885102, −0.11909148935263438302611734128,
1.45328027664496390013844445215, 2.53111625400919786979314569549, 3.76811134878409403669927393250, 5.09648352426601367728138485539, 5.71779144270973834385267761339, 6.77817557903788786991965276025, 7.25326902663462251343637050012, 8.309480884501549798829289713608, 8.706127813624109102124161778747, 9.615011811469517002214315951318
