| L(s) = 1 | − 6.72i·2-s − 8.31i·3-s − 29.1·4-s + 1.48i·5-s − 55.9·6-s − 13.7i·7-s + 88.6i·8-s + 11.7·9-s + 9.96·10-s − 10.2·11-s + 242. i·12-s + 98.4·13-s − 92.7·14-s + 12.3·15-s + 128.·16-s − 286.·17-s + ⋯ |
| L(s) = 1 | − 1.68i·2-s − 0.924i·3-s − 1.82·4-s + 0.0592i·5-s − 1.55·6-s − 0.281i·7-s + 1.38i·8-s + 0.145·9-s + 0.0996·10-s − 0.0843·11-s + 1.68i·12-s + 0.582·13-s − 0.473·14-s + 0.0548·15-s + 0.503·16-s − 0.991·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.357835 + 1.19851i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.357835 + 1.19851i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.54e3 + 1.01e3i)T \) |
| good | 2 | \( 1 + 6.72iT - 16T^{2} \) |
| 3 | \( 1 + 8.31iT - 81T^{2} \) |
| 5 | \( 1 - 1.48iT - 625T^{2} \) |
| 7 | \( 1 + 13.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 10.2T + 1.46e4T^{2} \) |
| 13 | \( 1 - 98.4T + 2.85e4T^{2} \) |
| 17 | \( 1 + 286.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 367. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 242.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.14e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 895.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.29e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.69e3T + 2.82e6T^{2} \) |
| 47 | \( 1 - 743.T + 4.87e6T^{2} \) |
| 53 | \( 1 - 99.3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.28e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 3.22e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 5.55e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.95e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 3.61e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 2.38e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 6.42e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 3.29e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.32e3T + 8.85e7T^{2} \) |
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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
−13.63409719513973738329664304696, −13.23966356583276260910784400502, −12.06748767970320461597982074378, −11.10776563197158773319785862571, −9.944989932053398330528920435852, −8.497208705492321835433251939389, −6.74629077503773298087293372423, −4.33869015343380834292318253105, −2.46503354508459468744653875224, −0.878613051298174907659235221261,
4.16893775230296432454822732375, 5.46627791896403259791043817940, 6.83959549429054552194910530726, 8.357888874946987861245732131176, 9.357884716054689167804915182441, 10.78493141643187849872303521215, 12.77426851382335095228552720984, 14.13597307783280004114507813875, 15.05471202278874009471985218741, 15.97215117054026921247706329140
