L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (1.23 + 1.86i)5-s + 0.999·6-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−2 − i)10-s + (−0.866 + 0.499i)12-s + (−4.33 − 2.5i)13-s + 0.999·14-s + (−0.133 − 2.23i)15-s + (−0.5 − 0.866i)16-s + (−1.73 + i)17-s + (−0.866 − 0.499i)18-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.550 + 0.834i)5-s + 0.408·6-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.632 − 0.316i)10-s + (−0.249 + 0.144i)12-s + (−1.20 − 0.693i)13-s + 0.267·14-s + (−0.0345 − 0.576i)15-s + (−0.125 − 0.216i)16-s + (−0.420 + 0.242i)17-s + (−0.204 − 0.117i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6952858210\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6952858210\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-1.23 - 1.86i)T \) |
| 37 | \( 1 + (6.06 + 0.5i)T \) |
good | 7 | \( 1 + (0.866 + 0.5i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (4.33 + 2.5i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + (-6.92 + 4i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 - 2i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.5 + 11.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.79 + 4.5i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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
−9.876874757016072366493189656406, −8.926905202943583591657720397426, −7.86961442498879737868250693892, −6.92950196837063429299856731171, −6.70429835637395341990676581612, −5.57459245676389861102001611281, −4.80145388149137668745289963673, −3.08898454717827627943658295329, −2.15238423174550775980865089708, −0.42296874529915027437676705718,
1.26225216188406648393603903523, 2.45706219617315935692470694742, 3.84615960783227203699126269995, 4.90193665456217271518080848471, 5.68151647113680921714901448052, 6.68245236504463031588586544681, 7.61524549700805570702717483146, 8.596325011281621855423577379957, 9.486884994826313470886945507352, 9.779074361977572562780972787165