| L(s) = 1 | + (−7.88 − 6.61i)2-s + (14.9 − 4.55i)3-s + (12.8 + 72.9i)4-s + (93.2 + 33.9i)5-s + (−147. − 62.7i)6-s + (−14.5 + 82.6i)7-s + (216. − 374. i)8-s + (201. − 135. i)9-s + (−510. − 884. i)10-s + (117. − 42.8i)11-s + (523. + 1.02e3i)12-s + (−345. + 290. i)13-s + (662. − 555. i)14-s + (1.54e3 + 81.3i)15-s + (−1.96e3 + 714. i)16-s + (11.5 + 19.9i)17-s + ⋯ |
| L(s) = 1 | + (−1.39 − 1.17i)2-s + (0.956 − 0.292i)3-s + (0.401 + 2.27i)4-s + (1.66 + 0.607i)5-s + (−1.67 − 0.711i)6-s + (−0.112 + 0.637i)7-s + (1.19 − 2.07i)8-s + (0.829 − 0.558i)9-s + (−1.61 − 2.79i)10-s + (0.293 − 0.106i)11-s + (1.05 + 2.06i)12-s + (−0.567 + 0.475i)13-s + (0.902 − 0.757i)14-s + (1.77 + 0.0933i)15-s + (−1.91 + 0.697i)16-s + (0.00967 + 0.0167i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.791i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.15115 - 0.566083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15115 - 0.566083i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-14.9 + 4.55i)T \) |
| good | 2 | \( 1 + (7.88 + 6.61i)T + (5.55 + 31.5i)T^{2} \) |
| 5 | \( 1 + (-93.2 - 33.9i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (14.5 - 82.6i)T + (-1.57e4 - 5.74e3i)T^{2} \) |
| 11 | \( 1 + (-117. + 42.8i)T + (1.23e5 - 1.03e5i)T^{2} \) |
| 13 | \( 1 + (345. - 290. i)T + (6.44e4 - 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-11.5 - 19.9i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-270. + 468. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (658. + 3.73e3i)T + (-6.04e6 + 2.20e6i)T^{2} \) |
| 29 | \( 1 + (1.36e3 + 1.14e3i)T + (3.56e6 + 2.01e7i)T^{2} \) |
| 31 | \( 1 + (-695. - 3.94e3i)T + (-2.69e7 + 9.79e6i)T^{2} \) |
| 37 | \( 1 + (3.83e3 + 6.63e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (1.00e4 - 8.44e3i)T + (2.01e7 - 1.14e8i)T^{2} \) |
| 43 | \( 1 + (1.67e4 - 6.10e3i)T + (1.12e8 - 9.44e7i)T^{2} \) |
| 47 | \( 1 + (-1.79e3 + 1.01e4i)T + (-2.15e8 - 7.84e7i)T^{2} \) |
| 53 | \( 1 - 5.89e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.54e4 - 5.63e3i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (1.04e3 - 5.90e3i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (4.84e3 - 4.06e3i)T + (2.34e8 - 1.32e9i)T^{2} \) |
| 71 | \( 1 + (3.41e4 + 5.91e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-7.43e3 + 1.28e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.74e4 + 3.14e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (2.19e4 + 1.84e4i)T + (6.84e8 + 3.87e9i)T^{2} \) |
| 89 | \( 1 + (7.06e4 - 1.22e5i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (6.07e4 - 2.21e4i)T + (6.57e9 - 5.51e9i)T^{2} \) |
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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
−16.71538422439960404054341153844, −14.64947997965568370983381731097, −13.38688768017316316336730280223, −12.13798822941547542828677495587, −10.38973631838266170097078522019, −9.525313853437776838292612227669, −8.654802291230207549085232333584, −6.80350977192108710388800663607, −2.80339977652186906691320835289, −1.82627498950313142818279512135,
1.57562485645252388851633863161, 5.45497053051900039337078118612, 7.13342401390065706872402443149, 8.553020100026915742637874337280, 9.700574597234482145969467260164, 10.12647907423992744881032614898, 13.36905448077936542090779011864, 14.26717708735934424980498581450, 15.47845831099031939554234209959, 16.80288781203300486718593430259
