L(s) = 1 | + (−1.88 + 0.656i)2-s − 1.73i·3-s + (3.13 − 2.48i)4-s + (1.13 + 3.27i)6-s + 9.55i·7-s + (−4.29 + 6.74i)8-s − 2.99·9-s − 9.92i·11-s + (−4.29 − 5.43i)12-s − 7.55·13-s + (−6.27 − 18.0i)14-s + (3.68 − 15.5i)16-s − 17.1·17-s + (5.66 − 1.97i)18-s + 26.1i·19-s + ⋯ |
L(s) = 1 | + (−0.944 + 0.328i)2-s − 0.577i·3-s + (0.784 − 0.620i)4-s + (0.189 + 0.545i)6-s + 1.36i·7-s + (−0.537 + 0.843i)8-s − 0.333·9-s − 0.902i·11-s + (−0.358 − 0.452i)12-s − 0.581·13-s + (−0.448 − 1.28i)14-s + (0.230 − 0.973i)16-s − 1.01·17-s + (0.314 − 0.109i)18-s + 1.37i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.207797 + 0.429258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.207797 + 0.429258i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.88 - 0.656i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 9.55iT - 49T^{2} \) |
| 11 | \( 1 + 9.92iT - 121T^{2} \) |
| 13 | \( 1 + 7.55T + 169T^{2} \) |
| 17 | \( 1 + 17.1T + 289T^{2} \) |
| 19 | \( 1 - 26.1iT - 361T^{2} \) |
| 23 | \( 1 - 1.67iT - 529T^{2} \) |
| 29 | \( 1 - 0.350T + 841T^{2} \) |
| 31 | \( 1 - 46.0iT - 961T^{2} \) |
| 37 | \( 1 + 22.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 77.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 41.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 14.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 22.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 94.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38T + 3.72e3T^{2} \) |
| 67 | \( 1 + 29.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 7.19iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 34.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 46.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 24.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 100.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 131.T + 9.40e3T^{2} \) |
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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
−11.84846841879249946625943480914, −10.88980205584520853523269673885, −9.809000596335805883322280803858, −8.674797135755125254267638197285, −8.362112495324022188063929049897, −7.05052622780103964456940202780, −6.11125356884304016042115949454, −5.29331146043923213967393318655, −2.93752835409112067014494226337, −1.71287557504209260346883154989,
0.29852556652790469363851623908, 2.23182389557541751073299093947, 3.78085566632352833159707844003, 4.80265910588590143078352346866, 6.75685691239396965785397080451, 7.29418121062184059719887867733, 8.497149791195841298286889317471, 9.535474993978305717557983383722, 10.16577895692111277779865468482, 10.96018310393876646462956501529