L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.14 + 1.98i)5-s + (−1.12 + 2.39i)7-s − 0.999·8-s − 2.29·10-s − 0.878·11-s + (−0.786 − 3.51i)13-s + (−2.63 + 0.222i)14-s + (−0.5 − 0.866i)16-s + (−3.20 + 5.54i)17-s + 1.50·19-s + (−1.14 − 1.98i)20-s + (−0.439 − 0.760i)22-s + (0.658 + 1.14i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.512 + 0.887i)5-s + (−0.425 + 0.904i)7-s − 0.353·8-s − 0.724·10-s − 0.264·11-s + (−0.218 − 0.975i)13-s + (−0.704 + 0.0593i)14-s + (−0.125 − 0.216i)16-s + (−0.777 + 1.34i)17-s + 0.346·19-s + (−0.256 − 0.443i)20-s + (−0.0936 − 0.162i)22-s + (0.137 + 0.237i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4615619553\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4615619553\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (1.12 - 2.39i)T \) |
| 13 | \( 1 + (0.786 + 3.51i)T \) |
good | 5 | \( 1 + (1.14 - 1.98i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 0.878T + 11T^{2} \) |
| 17 | \( 1 + (3.20 - 5.54i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 1.50T + 19T^{2} \) |
| 23 | \( 1 + (-0.658 - 1.14i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.669 + 1.15i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.94 + 3.37i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.69 + 8.12i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.80 - 3.12i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.95 + 8.58i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.188 + 0.327i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.22 - 2.11i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.98 - 5.16i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 4.81T + 61T^{2} \) |
| 67 | \( 1 - 9.75T + 67T^{2} \) |
| 71 | \( 1 + (1.02 + 1.77i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.432 - 0.749i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.18 - 7.24i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.66T + 83T^{2} \) |
| 89 | \( 1 + (6.41 + 11.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.40 - 7.62i)T + (-48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.922279398995192094098794916713, −8.891806815571275710195397737303, −8.255205598533723437227054737184, −7.41804777940943189090305889262, −6.73072174897674761746194992201, −5.85625185617381572993521548614, −5.26805091793364228200991556534, −3.95116089530870373609332517127, −3.23011744309070769780625940376, −2.26429670053348643629428087196,
0.15788906209981625401113486649, 1.38036062653787145110953187871, 2.78453657187582889643177857828, 3.78620171509677792321050629097, 4.68806429357893654540924267303, 5.05369955845632539053401495280, 6.56349280551938919130363886654, 7.09750930772882506200670119529, 8.183474679541196434050265585362, 8.988488950233110949636777982564