| L(s) = 1 | − 2.04i·2-s − 0.191·4-s − 2.23i·5-s + 4.50·7-s − 7.79i·8-s − 4.57·10-s + 3.31i·11-s + 19.7·13-s − 9.22i·14-s − 16.7·16-s − 14.3i·17-s + 3.92·19-s + 0.429i·20-s + 6.79·22-s + 12.0i·23-s + ⋯ |
| L(s) = 1 | − 1.02i·2-s − 0.0479·4-s − 0.447i·5-s + 0.643·7-s − 0.974i·8-s − 0.457·10-s + 0.301i·11-s + 1.51·13-s − 0.658i·14-s − 1.04·16-s − 0.846i·17-s + 0.206·19-s + 0.0214i·20-s + 0.308·22-s + 0.525i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.155074681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155074681\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 11 | \( 1 - 3.31iT \) |
| good | 2 | \( 1 + 2.04iT - 4T^{2} \) |
| 7 | \( 1 - 4.50T + 49T^{2} \) |
| 13 | \( 1 - 19.7T + 169T^{2} \) |
| 17 | \( 1 + 14.3iT - 289T^{2} \) |
| 19 | \( 1 - 3.92T + 361T^{2} \) |
| 23 | \( 1 - 12.0iT - 529T^{2} \) |
| 29 | \( 1 + 20.7iT - 841T^{2} \) |
| 31 | \( 1 + 2.93T + 961T^{2} \) |
| 37 | \( 1 - 18.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 39.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 25.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 18.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 6.60iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 50.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 73.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 53.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 53.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 97.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 47.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 151. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 35.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 137.T + 9.40e3T^{2} \) |
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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
−10.59831428549989805624230977965, −9.700140077204108087194284036832, −8.842831908037657476415383332136, −7.83824057511567090746781446238, −6.76116551348154062787348879494, −5.58684645997463049588478446282, −4.37650123770746407590096270543, −3.38900184340020415024752332549, −2.01841185732389753769369382056, −0.947103114153690517427111992815,
1.62520918495104702136900817737, 3.19464432874590082663791750410, 4.56028561890779305940892794111, 5.81051255358906283800850238177, 6.35212480812680383683554790846, 7.37572014412140262535811845458, 8.277425002243117110325414122048, 8.787961725168325971757713824767, 10.30705744314500904157011700428, 11.10031734060870638863462063873
