L(s) = 1 | + (−0.618 + 1.61i)3-s + (−2.46 + 1.42i)5-s + (1.02 + 2.44i)7-s + (−2.23 − 2.00i)9-s + (−2.42 + 4.20i)11-s + 2.75·13-s + (−0.780 − 4.87i)15-s + (1.75 − 3.03i)17-s + (−3.14 − 5.44i)19-s + (−4.58 + 0.142i)21-s + (−3.15 + 1.82i)23-s + (1.56 − 2.71i)25-s + (4.61 − 2.38i)27-s − 3.90·29-s + (0.858 + 0.495i)31-s + ⋯ |
L(s) = 1 | + (−0.356 + 0.934i)3-s + (−1.10 + 0.637i)5-s + (0.385 + 0.922i)7-s + (−0.745 − 0.666i)9-s + (−0.731 + 1.26i)11-s + 0.763·13-s + (−0.201 − 1.25i)15-s + (0.425 − 0.736i)17-s + (−0.721 − 1.24i)19-s + (−0.999 + 0.0309i)21-s + (−0.658 + 0.380i)23-s + (0.313 − 0.542i)25-s + (0.888 − 0.458i)27-s − 0.725·29-s + (0.154 + 0.0889i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.129884 - 0.453659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.129884 - 0.453659i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.618 - 1.61i)T \) |
| 7 | \( 1 + (-1.02 - 2.44i)T \) |
good | 5 | \( 1 + (2.46 - 1.42i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.42 - 4.20i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 + (-1.75 + 3.03i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.14 + 5.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.15 - 1.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 + (-0.858 - 0.495i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.06 + 0.614i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.10T + 41T^{2} \) |
| 43 | \( 1 - 5.11iT - 43T^{2} \) |
| 47 | \( 1 + (5.61 + 9.72i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.00 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.890 + 0.514i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.24 - 2.15i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.02 + 2.90i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.75iT - 71T^{2} \) |
| 73 | \( 1 + (-0.291 - 0.168i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.80 - 4.85i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.138iT - 83T^{2} \) |
| 89 | \( 1 + (-0.580 - 1.00i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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
−11.15564547446629112226221649392, −10.22694688728609306228415834406, −9.375312514019760807734643937104, −8.454581279392354389134055661582, −7.58099764248128730559533246837, −6.58459314373568467129242961170, −5.37570454873483477843447673738, −4.62002993083649834195922665432, −3.59303063714479241276741597601, −2.47193237628440755163408969091,
0.26778282190097544423184347893, 1.52409291326861942160936890197, 3.42313925838864565449589679101, 4.32235326424593924306904903071, 5.61521538523150300724417039939, 6.37119149868040547831971546435, 7.73265957731000905177620955645, 8.037607380327700394250876804071, 8.632739823251536943701642315392, 10.39163926191693160202118197817