L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.724 + 1.57i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−1.72 + 0.158i)6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−1.94 − 2.28i)9-s − 0.999·10-s + (0.275 + 0.476i)11-s + (−1 − 1.41i)12-s + (−2.22 + 3.85i)13-s + (0.499 − 0.866i)14-s + (−1 − 1.41i)15-s + (−0.5 − 0.866i)16-s − 3·17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.418 + 0.908i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.704 + 0.0648i)6-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (−0.649 − 0.760i)9-s − 0.316·10-s + (0.0829 + 0.143i)11-s + (−0.288 − 0.408i)12-s + (−0.617 + 1.06i)13-s + (0.133 − 0.231i)14-s + (−0.258 − 0.365i)15-s + (−0.125 − 0.216i)16-s − 0.727·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.241558 - 0.511843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.241558 - 0.511843i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.724 - 1.57i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 11 | \( 1 + (-0.275 - 0.476i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.22 - 3.85i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 5.89T + 19T^{2} \) |
| 23 | \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 + (-5.72 + 9.91i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.94 - 5.10i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.89 - 8.48i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 + (3.94 - 6.84i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.67 - 4.63i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.84 - 11.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.44T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.55 - 6.14i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + (-7.84 - 13.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.02855909305763266724105695831, −10.39493867417529589322236414126, −9.299132926260239378151538291047, −8.711790873925217765685638466973, −7.35732363653874391606270715404, −6.61568588896758886937402658648, −5.78555221685826586918175949462, −4.39409358381716813073097864783, −4.18095450002394439835587017002, −2.64026001773433143198087097211,
0.27429283714470285503377670198, 1.89658907327213209975539413863, 3.01028986725345669538049307976, 4.47850422501264974922527697330, 5.45293387744374220769990100715, 6.26362422706255571231920052194, 7.34508524650094070792123615886, 8.318259508424400016053931137350, 9.122916231389387713930640220282, 10.33302794347077239239209962606