| L(s) = 1 | + (−2.23 + 1.46i)2-s + (2.03 − 4.70i)4-s + (−1.73 + 1.84i)5-s + (−0.507 − 0.0593i)7-s + (1.44 + 8.22i)8-s + (1.17 − 6.66i)10-s + (0.440 − 1.47i)11-s + (−0.328 + 5.63i)13-s + (1.21 − 0.612i)14-s + (−8.25 − 8.75i)16-s + (1.33 − 0.484i)17-s + (0.986 + 0.359i)19-s + (5.14 + 11.9i)20-s + (1.17 + 3.93i)22-s + (0.258 − 0.0302i)23-s + ⋯ |
| L(s) = 1 | + (−1.57 + 1.03i)2-s + (1.01 − 2.35i)4-s + (−0.778 + 0.824i)5-s + (−0.191 − 0.0224i)7-s + (0.512 + 2.90i)8-s + (0.371 − 2.10i)10-s + (0.132 − 0.443i)11-s + (−0.0909 + 1.56i)13-s + (0.325 − 0.163i)14-s + (−2.06 − 2.18i)16-s + (0.322 − 0.117i)17-s + (0.226 + 0.0824i)19-s + (1.15 + 2.66i)20-s + (0.250 + 0.838i)22-s + (0.0539 − 0.00630i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0748951 - 0.129069i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0748951 - 0.129069i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.23 - 1.46i)T + (0.792 - 1.83i)T^{2} \) |
| 5 | \( 1 + (1.73 - 1.84i)T + (-0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (0.507 + 0.0593i)T + (6.81 + 1.61i)T^{2} \) |
| 11 | \( 1 + (-0.440 + 1.47i)T + (-9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (0.328 - 5.63i)T + (-12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (-1.33 + 0.484i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.986 - 0.359i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.258 + 0.0302i)T + (22.3 - 5.30i)T^{2} \) |
| 29 | \( 1 + (1.58 + 0.798i)T + (17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (-3.08 - 4.14i)T + (-8.89 + 29.6i)T^{2} \) |
| 37 | \( 1 + (5.04 + 4.23i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (5.64 + 3.71i)T + (16.2 + 37.6i)T^{2} \) |
| 43 | \( 1 + (5.94 - 1.40i)T + (38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (4.36 - 5.86i)T + (-13.4 - 45.0i)T^{2} \) |
| 53 | \( 1 + (4.74 - 8.21i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.42 + 4.74i)T + (-49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (5.10 + 11.8i)T + (-41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (-5.73 + 2.88i)T + (40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (-0.896 + 5.08i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.03 + 5.87i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (8.30 - 5.46i)T + (31.2 - 72.5i)T^{2} \) |
| 83 | \( 1 + (-10.8 + 7.11i)T + (32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (-1.71 - 9.71i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.46 - 2.61i)T + (-5.64 + 96.8i)T^{2} \) |
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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
−10.82589895838346129550720043453, −9.814605780494960983067565003872, −9.177058452396577205537443966506, −8.310177199421821957272348944743, −7.52494362732702840257870198090, −6.79662457868438826946856045184, −6.27732935294989702174427371499, −4.94038389588254344371358277404, −3.37403273840630881335093264379, −1.70033658077695437030860698661,
0.13901428513054495104522601325, 1.36216705085491471676011953351, 2.87045929127098586166292593072, 3.77281229448013390025330877820, 5.09738113545831989000584112714, 6.74332789266753302276582747223, 7.84504533803467396507250600177, 8.173073646726966825552730888466, 8.994594552074116040603947371903, 9.991516096432953975981695639579
