L(s) = 1 | + (0.987 + 0.156i)2-s + (0.513 − 1.65i)3-s + (0.951 + 0.309i)4-s + (−0.545 − 2.16i)5-s + (0.765 − 1.55i)6-s + (−1.41 + 1.41i)7-s + (0.891 + 0.453i)8-s + (−2.47 − 1.69i)9-s + (−0.199 − 2.22i)10-s + (2.37 + 3.26i)11-s + (0.999 − 1.41i)12-s + (0.301 + 1.90i)13-s + (−1.62 + 1.17i)14-s + (−3.86 − 0.211i)15-s + (0.809 + 0.587i)16-s + (−1.78 + 3.50i)17-s + ⋯ |
L(s) = 1 | + (0.698 + 0.110i)2-s + (0.296 − 0.955i)3-s + (0.475 + 0.154i)4-s + (−0.243 − 0.969i)5-s + (0.312 − 0.634i)6-s + (−0.535 + 0.535i)7-s + (0.315 + 0.160i)8-s + (−0.824 − 0.566i)9-s + (−0.0629 − 0.704i)10-s + (0.715 + 0.984i)11-s + (0.288 − 0.408i)12-s + (0.0836 + 0.528i)13-s + (−0.433 + 0.314i)14-s + (−0.998 − 0.0546i)15-s + (0.202 + 0.146i)16-s + (−0.432 + 0.849i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.51211 - 0.623616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.51211 - 0.623616i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.987 - 0.156i)T \) |
| 3 | \( 1 + (-0.513 + 1.65i)T \) |
| 5 | \( 1 + (0.545 + 2.16i)T \) |
good | 7 | \( 1 + (1.41 - 1.41i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.37 - 3.26i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.301 - 1.90i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (1.78 - 3.50i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-6.42 + 2.08i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.317 + 2.00i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (1.79 - 5.53i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (2.88 + 8.86i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (6.31 - 1.00i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (0.756 - 1.04i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (6.68 + 6.68i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.18 + 0.601i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (4.96 + 9.73i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-3.81 - 2.77i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.433 - 0.315i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.35 + 0.688i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (0.520 + 0.169i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.37 - 1.16i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-10.2 - 3.31i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.00 - 1.53i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-11.4 + 8.34i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (4.82 + 9.47i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−12.84870191507222467095060973299, −12.19609747512430418664428010977, −11.49077178947990567138716100749, −9.512376190940400955952340553364, −8.691529405415249007606274413053, −7.40811993040777255410383086565, −6.45093674694979768464580461476, −5.18713170596810053671383299595, −3.68704341457485041195919784083, −1.86182122855525303998286444469,
3.12974096959592712105737293623, 3.68164453997796656759543431255, 5.30280474815719736940457459536, 6.55398238712791465998323139374, 7.76781893950029266514395971413, 9.303167109795606340855240704819, 10.28435235751776523402050533027, 11.12574353264594802743648794818, 11.90634514467916657785776961217, 13.67652687516166661351978269452