L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)4-s + (−0.690 + 2.12i)5-s + 4.47·7-s + (0.927 − 2.85i)8-s + (−1.80 + 1.31i)10-s + (2.61 + 1.90i)11-s + (−2.73 + 1.98i)13-s + (3.61 + 2.62i)14-s + (0.809 − 0.587i)16-s + (0.881 − 2.71i)17-s + (−1 + 3.07i)19-s + 2.23·20-s + (1 + 3.07i)22-s + (−3.61 − 2.62i)23-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.154 − 0.475i)4-s + (−0.309 + 0.951i)5-s + 1.69·7-s + (0.327 − 1.00i)8-s + (−0.572 + 0.415i)10-s + (0.789 + 0.573i)11-s + (−0.758 + 0.551i)13-s + (0.966 + 0.702i)14-s + (0.202 − 0.146i)16-s + (0.213 − 0.658i)17-s + (−0.229 + 0.706i)19-s + 0.499·20-s + (0.213 + 0.656i)22-s + (−0.754 − 0.548i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.63552 + 0.419930i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63552 + 0.419930i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.690 - 2.12i)T \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T + (0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + (-2.61 - 1.90i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.73 - 1.98i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.881 + 2.71i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1 - 3.07i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (3.61 + 2.62i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.35 + 4.16i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.23 + 6.88i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (6.54 - 4.75i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (1.11 - 0.812i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + (-1.61 - 4.97i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.427 + 1.31i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.23 - 2.35i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.363i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.61 + 4.97i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.236 - 0.726i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.5 + 1.81i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.09 - 3.35i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.16 - 4.47i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.73 - 8.42i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−12.08851342813807354937723965215, −11.54234059217981358406644650788, −10.41628445832067448545366468273, −9.584383007897789083075582476923, −8.058179543206038949223117324633, −7.17897739639323865238545785313, −6.17268030779864550328744546749, −4.84822832189948904650503146037, −4.08437446134245513321465711808, −1.93949826677312113901839068846,
1.72749929965010951357273397756, 3.62989396027649384536275826280, 4.71824789779198342601064649634, 5.39051016927796325163416647458, 7.42790405120259009925472471808, 8.349707992381493234343423797379, 8.860013480804966148367741146245, 10.58407970202855177728717922052, 11.60235054872620918046424035094, 12.05632783391819231813050025346