| L(s) = 1 | − 1.57·2-s + 0.491·4-s + 1.67·5-s + 2.77·7-s + 2.38·8-s − 2.64·10-s − 4.15·11-s + 6.87·13-s − 4.38·14-s − 4.74·16-s + 0.976·17-s + 2.68·19-s + 0.824·20-s + 6.55·22-s + 1.61·23-s − 2.18·25-s − 10.8·26-s + 1.36·28-s − 8.22·29-s + 1.04·31-s + 2.72·32-s − 1.54·34-s + 4.66·35-s − 1.30·37-s − 4.23·38-s + 3.99·40-s − 4.84·41-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.245·4-s + 0.750·5-s + 1.05·7-s + 0.841·8-s − 0.837·10-s − 1.25·11-s + 1.90·13-s − 1.17·14-s − 1.18·16-s + 0.236·17-s + 0.616·19-s + 0.184·20-s + 1.39·22-s + 0.336·23-s − 0.436·25-s − 2.12·26-s + 0.258·28-s − 1.52·29-s + 0.187·31-s + 0.481·32-s − 0.264·34-s + 0.788·35-s − 0.215·37-s − 0.687·38-s + 0.631·40-s − 0.757·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059222498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059222498\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.57T + 2T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 - 6.87T + 13T^{2} \) |
| 17 | \( 1 - 0.976T + 17T^{2} \) |
| 19 | \( 1 - 2.68T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 + 8.22T + 29T^{2} \) |
| 31 | \( 1 - 1.04T + 31T^{2} \) |
| 37 | \( 1 + 1.30T + 37T^{2} \) |
| 41 | \( 1 + 4.84T + 41T^{2} \) |
| 43 | \( 1 - 9.84T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 - 9.05T + 59T^{2} \) |
| 61 | \( 1 + 1.28T + 61T^{2} \) |
| 67 | \( 1 + 4.64T + 67T^{2} \) |
| 71 | \( 1 - 5.62T + 71T^{2} \) |
| 73 | \( 1 + 4.56T + 73T^{2} \) |
| 79 | \( 1 - 4.65T + 79T^{2} \) |
| 83 | \( 1 - 5.76T + 83T^{2} \) |
| 89 | \( 1 - 4.54T + 89T^{2} \) |
| 97 | \( 1 - 8.57T + 97T^{2} \) |
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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
−10.45932299482688333501306776339, −9.411206781486181995646817152710, −8.719342320152038539880779646738, −7.983455581894657076721036042878, −7.31838791955465934951377609864, −5.86091592589307152553643596876, −5.19378731119956203054696392547, −3.87763835950422610210172208167, −2.17405774737837463375929275576, −1.11744055592954130864401262748,
1.11744055592954130864401262748, 2.17405774737837463375929275576, 3.87763835950422610210172208167, 5.19378731119956203054696392547, 5.86091592589307152553643596876, 7.31838791955465934951377609864, 7.983455581894657076721036042878, 8.719342320152038539880779646738, 9.411206781486181995646817152710, 10.45932299482688333501306776339
