| L(s) = 1 | + (−0.730 − 0.266i)2-s + (−1.06 − 0.896i)4-s + (0.412 − 2.34i)5-s + (−1.91 + 1.60i)7-s + (1.32 + 2.28i)8-s + (−0.924 + 1.60i)10-s + (0.545 + 3.09i)11-s + (−1.25 + 0.457i)13-s + (1.82 − 0.665i)14-s + (0.127 + 0.725i)16-s + (−3.13 + 5.43i)17-s + (−4.03 − 6.98i)19-s + (−2.53 + 2.13i)20-s + (0.424 − 2.40i)22-s + (3.10 + 2.60i)23-s + ⋯ |
| L(s) = 1 | + (−0.516 − 0.188i)2-s + (−0.534 − 0.448i)4-s + (0.184 − 1.04i)5-s + (−0.724 + 0.607i)7-s + (0.466 + 0.808i)8-s + (−0.292 + 0.506i)10-s + (0.164 + 0.932i)11-s + (−0.348 + 0.126i)13-s + (0.488 − 0.177i)14-s + (0.0319 + 0.181i)16-s + (−0.760 + 1.31i)17-s + (−0.925 − 1.60i)19-s + (−0.567 + 0.476i)20-s + (0.0904 − 0.512i)22-s + (0.647 + 0.543i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.560059 + 0.281272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.560059 + 0.281272i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.730 + 0.266i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.412 + 2.34i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.91 - 1.60i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.545 - 3.09i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (1.25 - 0.457i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.13 - 5.43i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.03 + 6.98i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.10 - 2.60i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-8.72 - 3.17i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.16 - 1.82i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (2.76 - 4.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.67 + 2.43i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.405 - 2.30i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (3.53 - 2.96i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 0.135T + 53T^{2} \) |
| 59 | \( 1 + (0.694 - 3.93i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.261 + 0.219i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (9.51 - 3.46i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (4.09 - 7.09i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.15 - 10.6i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.83 + 1.39i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.858 + 0.312i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-1.86 - 3.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.04 + 5.90i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.29561340268701925537771210453, −9.517242481562749413873802645797, −8.832836492200927983154804406435, −8.501811454664952951556822631063, −6.99021163418058984152807193647, −6.06901079421703887708844124477, −4.87861225326813497645859850432, −4.45825288084052972659872001939, −2.55728499087470457597795258589, −1.28332672869212795329662540436,
0.42964314180149260454168666150, 2.71549873705370943526757128435, 3.59829758551640076492950975899, 4.63714069067004768812527836282, 6.22843475266527430860714384367, 6.78879269436789340719499250193, 7.67779015967810796444228798762, 8.559256204241471492899157091446, 9.429250684385576779044202899661, 10.27066065393593114606202895077
