L(s) = 1 | + (−1.92 + 0.545i)2-s − 4.54i·3-s + (3.40 − 2.10i)4-s + 1.36·5-s + (2.48 + 8.74i)6-s − 2.64i·7-s + (−5.40 + 5.89i)8-s − 11.6·9-s + (−2.63 + 0.746i)10-s + 15.3i·11-s + (−9.54 − 15.4i)12-s + 19.9·13-s + (1.44 + 5.09i)14-s − 6.21i·15-s + (7.17 − 14.2i)16-s − 4.73·17-s + ⋯ |
L(s) = 1 | + (−0.962 + 0.272i)2-s − 1.51i·3-s + (0.851 − 0.525i)4-s + 0.273·5-s + (0.413 + 1.45i)6-s − 0.377i·7-s + (−0.675 + 0.737i)8-s − 1.29·9-s + (−0.263 + 0.0746i)10-s + 1.39i·11-s + (−0.795 − 1.28i)12-s + 1.53·13-s + (0.103 + 0.363i)14-s − 0.414i·15-s + (0.448 − 0.893i)16-s − 0.278·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.525 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.602071 - 0.336011i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.602071 - 0.336011i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.92 - 0.545i)T \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 + 4.54iT - 9T^{2} \) |
| 5 | \( 1 - 1.36T + 25T^{2} \) |
| 11 | \( 1 - 15.3iT - 121T^{2} \) |
| 13 | \( 1 - 19.9T + 169T^{2} \) |
| 17 | \( 1 + 4.73T + 289T^{2} \) |
| 19 | \( 1 - 13.2iT - 361T^{2} \) |
| 23 | \( 1 + 14.9iT - 529T^{2} \) |
| 29 | \( 1 - 6.20T + 841T^{2} \) |
| 31 | \( 1 - 18.5iT - 961T^{2} \) |
| 37 | \( 1 + 27.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 27.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 30.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 12.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 28.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 6.52T + 3.72e3T^{2} \) |
| 67 | \( 1 - 102. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 45.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 70.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 33.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 159. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 50.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 89.7T + 9.40e3T^{2} \) |
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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
−17.34439056618848207821870331104, −15.86274734450940922703739786063, −14.31253467648419788916335462958, −13.03288612571902477814342810913, −11.77317084631431915866697327411, −10.23174790777163133091244438011, −8.507554387749978695081078152812, −7.28675889816114551751227062113, −6.22775598100866860173054403083, −1.68304587108463664026036144013,
3.50379472561579913845785116931, 5.94205506762712360266347020754, 8.494284247850643041540154761718, 9.359917243644393456362464360202, 10.70228510988451336569799943121, 11.43870571015080607516480459093, 13.60061345888721301558260892673, 15.47518231717247330750589008081, 15.97954147200388943795178546990, 17.07050026521147063909929678090