L(s) = 1 | + (256 − 256i)2-s + (2.59e4 + 2.59e4i)3-s − 1.31e5i·4-s + (−1.76e6 + 8.32e5i)5-s + 1.33e7·6-s + (−2.67e7 + 2.67e7i)7-s + (−3.35e7 − 3.35e7i)8-s + 9.62e8i·9-s + (−2.39e8 + 6.65e8i)10-s − 7.63e8·11-s + (3.40e9 − 3.40e9i)12-s + (−1.08e10 − 1.08e10i)13-s + 1.36e10i·14-s + (−6.75e10 − 2.42e10i)15-s − 1.71e10·16-s + (4.48e10 − 4.48e10i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (1.31 + 1.31i)3-s − 0.5i·4-s + (−0.904 + 0.426i)5-s + 1.31·6-s + (−0.662 + 0.662i)7-s + (−0.250 − 0.250i)8-s + 2.48i·9-s + (−0.239 + 0.665i)10-s − 0.323·11-s + (0.659 − 0.659i)12-s + (−1.02 − 1.02i)13-s + 0.662i·14-s + (−1.75 − 0.631i)15-s − 0.250·16-s + (0.378 − 0.378i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(0.944650 + 1.95871i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.944650 + 1.95871i\) |
\(L(10)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-256 + 256i)T \) |
| 5 | \( 1 + (1.76e6 - 8.32e5i)T \) |
good | 3 | \( 1 + (-2.59e4 - 2.59e4i)T + 3.87e8iT^{2} \) |
| 7 | \( 1 + (2.67e7 - 2.67e7i)T - 1.62e15iT^{2} \) |
| 11 | \( 1 + 7.63e8T + 5.55e18T^{2} \) |
| 13 | \( 1 + (1.08e10 + 1.08e10i)T + 1.12e20iT^{2} \) |
| 17 | \( 1 + (-4.48e10 + 4.48e10i)T - 1.40e22iT^{2} \) |
| 19 | \( 1 - 3.49e11iT - 1.04e23T^{2} \) |
| 23 | \( 1 + (-9.76e11 - 9.76e11i)T + 3.24e24iT^{2} \) |
| 29 | \( 1 - 2.06e13iT - 2.10e26T^{2} \) |
| 31 | \( 1 - 1.69e13T + 6.99e26T^{2} \) |
| 37 | \( 1 + (9.71e13 - 9.71e13i)T - 1.68e28iT^{2} \) |
| 41 | \( 1 - 2.27e14T + 1.07e29T^{2} \) |
| 43 | \( 1 + (-5.68e14 - 5.68e14i)T + 2.52e29iT^{2} \) |
| 47 | \( 1 + (-1.52e14 + 1.52e14i)T - 1.25e30iT^{2} \) |
| 53 | \( 1 + (1.38e15 + 1.38e15i)T + 1.08e31iT^{2} \) |
| 59 | \( 1 + 9.12e15iT - 7.50e31T^{2} \) |
| 61 | \( 1 + 9.20e15T + 1.36e32T^{2} \) |
| 67 | \( 1 + (-2.50e16 + 2.50e16i)T - 7.40e32iT^{2} \) |
| 71 | \( 1 - 7.31e15T + 2.10e33T^{2} \) |
| 73 | \( 1 + (-1.46e16 - 1.46e16i)T + 3.46e33iT^{2} \) |
| 79 | \( 1 + 1.58e17iT - 1.43e34T^{2} \) |
| 83 | \( 1 + (-1.61e17 - 1.61e17i)T + 3.49e34iT^{2} \) |
| 89 | \( 1 - 2.29e17iT - 1.22e35T^{2} \) |
| 97 | \( 1 + (-1.30e17 + 1.30e17i)T - 5.77e35iT^{2} \) |
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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
−16.05578731729545677488834331707, −15.21825381338190031475688728345, −14.32837629755935066739764750527, −12.51666096841470098033456782445, −10.62190564694141800145402745237, −9.514419311890539150146933290998, −7.903797661440814543397003795267, −5.00625333875180864627312012222, −3.41897381617851833997536920240, −2.76535467909683955626955859506,
0.54145104970842437376329615526, 2.59055197177251470073576155211, 4.07698888876322726180031619577, 6.85613304245188063366321471736, 7.66408975026996835606947549092, 9.047748620311745585500490344506, 12.11800827256218814217576704062, 13.09990048453661645739808792946, 14.18870930096609929931985205702, 15.47117396430951385971223086314