L(s) = 1 | + (1.44 − 0.253i)2-s + (−0.970 − 1.15i)3-s + (0.130 − 0.0473i)4-s + (−1.69 − 1.41i)6-s + (−3.52 − 2.03i)7-s + (−2.35 + 1.36i)8-s + (0.124 − 0.707i)9-s + (0.310 + 0.537i)11-s + (−0.180 − 0.104i)12-s + (−3.27 + 3.90i)13-s + (−5.59 − 2.03i)14-s + (−3.26 + 2.73i)16-s + (0.262 − 0.0462i)17-s − 1.05i·18-s + (−0.399 − 4.34i)19-s + ⋯ |
L(s) = 1 | + (1.01 − 0.179i)2-s + (−0.560 − 0.667i)3-s + (0.0650 − 0.0236i)4-s + (−0.690 − 0.579i)6-s + (−1.33 − 0.769i)7-s + (−0.833 + 0.481i)8-s + (0.0416 − 0.235i)9-s + (0.0936 + 0.162i)11-s + (−0.0522 − 0.0301i)12-s + (−0.908 + 1.08i)13-s + (−1.49 − 0.544i)14-s + (−0.815 + 0.684i)16-s + (0.0636 − 0.0112i)17-s − 0.247i·18-s + (−0.0917 − 0.995i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00204404 + 0.593349i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00204404 + 0.593349i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (0.399 + 4.34i)T \) |
good | 2 | \( 1 + (-1.44 + 0.253i)T + (1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (0.970 + 1.15i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (3.52 + 2.03i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.310 - 0.537i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.27 - 3.90i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.262 + 0.0462i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (1.99 + 5.48i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.708 - 4.01i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.83iT - 37T^{2} \) |
| 41 | \( 1 + (3.43 - 2.88i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.615 + 1.69i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-11.3 - 2.00i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (0.862 + 2.37i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.154 + 0.876i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.03 - 0.742i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (13.8 + 2.44i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.54 - 0.563i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (1.15 + 1.37i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (3.94 - 3.30i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-11.9 - 6.89i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.000572 - 0.000480i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-10.2 + 1.80i)T + (91.1 - 33.1i)T^{2} \) |
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
−10.81940307426460205774239797540, −9.633745740462703521650555581837, −8.994818001159293123429829522883, −7.31616147475638644220674549325, −6.67190171440743742930595051637, −5.94708555682220302306935916970, −4.61419975250112164417309952745, −3.80664067067790972099937748494, −2.51699871289661178111296078785, −0.26369108647101514758201053163,
2.83586228671926723115904975072, 3.76839378354001180188609026370, 4.96394112413613867825449175556, 5.68364208831406602874941436346, 6.28133090821737653320652541018, 7.66947829093748218359817661504, 9.000335083124350274634142876479, 9.947072808109142706254344567306, 10.30115680644584277285690053247, 11.88878892481190501928443765941