L(s) = 1 | + 4.24i·2-s − 1.99·4-s − 29.6i·5-s − 28·7-s + 59.3i·8-s + 125.·10-s + 16.9i·11-s − 112·13-s − 118. i·14-s − 284·16-s − 89.0i·17-s + 560·19-s + 59.3i·20-s − 71.9·22-s + 797. i·23-s + ⋯ |
L(s) = 1 | + 1.06i·2-s − 0.124·4-s − 1.18i·5-s − 0.571·7-s + 0.928i·8-s + 1.25·10-s + 0.140i·11-s − 0.662·13-s − 0.606i·14-s − 1.10·16-s − 0.308i·17-s + 1.55·19-s + 0.148i·20-s − 0.148·22-s + 1.50i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.907074 + 0.469536i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.907074 + 0.469536i\) |
\(L(3)\) |
|
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 | \( 1 - 4.24iT - 16T^{2} \) |
| 5 | \( 1 + 29.6iT - 625T^{2} \) |
| 7 | \( 1 + 28T + 2.40e3T^{2} \) |
| 11 | \( 1 - 16.9iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 112T + 2.85e4T^{2} \) |
| 17 | \( 1 + 89.0iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 560T + 1.30e5T^{2} \) |
| 23 | \( 1 - 797. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 988. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 364T + 9.23e5T^{2} \) |
| 37 | \( 1 + 826T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.81e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.73e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.30e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.79e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 4.51e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.61e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 3.78e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 8.60e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.60e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.27e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 118. iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 4.36e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 5.82e3T + 8.85e7T^{2} \) |
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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
−20.79834550994884979452667231040, −19.68945706363571796124802158745, −17.56178094444797750077449670346, −16.45086328334429934111289160151, −15.51072000936153803521639759887, −13.69219155187576034968971006764, −11.97386217800563039958632606030, −9.324655210416613825846300216378, −7.50214236652905414179581649637, −5.39176983631017012344750806467,
3.02878905212755916404233430378, 6.87871834439669389065105973666, 9.880286965419803764732677726567, 11.08512464056210486461730968623, 12.58650245369950066745516405039, 14.43674555261655475358626398397, 16.11414509296955556598322522095, 18.24221438119862436751232998949, 19.23754369912055127718612017883, 20.39814851515352733791280225936