| L(s) = 1 | + (−1.62 − 1.36i)2-s + (0.436 + 2.47i)4-s + (1.94 + 0.708i)5-s + (0.841 − 4.77i)7-s + (0.547 − 0.949i)8-s + (−2.20 − 3.81i)10-s + (−3.89 + 1.41i)11-s + (−0.931 + 0.781i)13-s + (−7.89 + 6.62i)14-s + (2.53 − 0.924i)16-s + (−1.18 − 2.04i)17-s + (0.919 − 1.59i)19-s + (−0.904 + 5.13i)20-s + (8.28 + 3.01i)22-s + (−0.747 − 4.23i)23-s + ⋯ |
| L(s) = 1 | + (−1.15 − 0.965i)2-s + (0.218 + 1.23i)4-s + (0.870 + 0.316i)5-s + (0.318 − 1.80i)7-s + (0.193 − 0.335i)8-s + (−0.695 − 1.20i)10-s + (−1.17 + 0.427i)11-s + (−0.258 + 0.216i)13-s + (−2.10 + 1.77i)14-s + (0.634 − 0.231i)16-s + (−0.286 − 0.496i)17-s + (0.210 − 0.365i)19-s + (−0.202 + 1.14i)20-s + (1.76 + 0.642i)22-s + (−0.155 − 0.883i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0377385 - 0.647945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0377385 - 0.647945i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.62 + 1.36i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-1.94 - 0.708i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.841 + 4.77i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (3.89 - 1.41i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.931 - 0.781i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.18 + 2.04i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.919 + 1.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.747 + 4.23i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.28 - 1.91i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.255 - 1.45i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (4.48 + 7.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.73 + 1.45i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.15 + 1.87i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.24 + 7.07i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + (0.246 + 0.0896i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.773 - 4.38i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.16 + 2.65i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.54 + 2.67i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.38 - 11.0i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.48 - 2.92i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.47 + 5.43i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (8.48 - 14.6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.80 + 1.74i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.17592261836784970836961135286, −9.476830756582052017464804955929, −8.421933016176346380884952162050, −7.49153326810801626657855049981, −6.89556545176258099906813280677, −5.38354390184831735762426395815, −4.25041895048044688746810906203, −2.83170459323646686676324940354, −1.88612942088287916686372951353, −0.47482208794464278740935962647,
1.70149557297191593032543342730, 2.90446686712649650751191926120, 5.07050110358790078372625537769, 5.77136539848179671956095052659, 6.24060565356262178749155060105, 7.71623553710059514524835907024, 8.231265631056569464409791699684, 9.022561544791193190059808011700, 9.607367769169231515143703565430, 10.37666291627938600448051199947
