L(s) = 1 | + 0.634i·2-s + (1.99 + 2.23i)3-s + 3.59·4-s − 4.18i·5-s + (−1.42 + 1.26i)6-s − 4.15·7-s + 4.82i·8-s + (−1.02 + 8.94i)9-s + 2.65·10-s − 2.77i·11-s + (7.18 + 8.05i)12-s − 4.48·13-s − 2.63i·14-s + (9.36 − 8.35i)15-s + 11.3·16-s − 14.7i·17-s + ⋯ |
L(s) = 1 | + 0.317i·2-s + (0.665 + 0.746i)3-s + 0.899·4-s − 0.836i·5-s + (−0.236 + 0.211i)6-s − 0.594·7-s + 0.602i·8-s + (−0.113 + 0.993i)9-s + 0.265·10-s − 0.251i·11-s + (0.598 + 0.671i)12-s − 0.344·13-s − 0.188i·14-s + (0.624 − 0.556i)15-s + 0.708·16-s − 0.869i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.50919 + 0.575369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50919 + 0.575369i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.99 - 2.23i)T \) |
| 23 | \( 1 - 4.79iT \) |
good | 2 | \( 1 - 0.634iT - 4T^{2} \) |
| 5 | \( 1 + 4.18iT - 25T^{2} \) |
| 7 | \( 1 + 4.15T + 49T^{2} \) |
| 11 | \( 1 + 2.77iT - 121T^{2} \) |
| 13 | \( 1 + 4.48T + 169T^{2} \) |
| 17 | \( 1 + 14.7iT - 289T^{2} \) |
| 19 | \( 1 + 32.7T + 361T^{2} \) |
| 29 | \( 1 + 52.7iT - 841T^{2} \) |
| 31 | \( 1 - 17.7T + 961T^{2} \) |
| 37 | \( 1 - 16.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 5.53iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 9.31T + 1.84e3T^{2} \) |
| 47 | \( 1 - 56.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 103. iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 53.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 41.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 33.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 51.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 34.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 100.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 57.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 92.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 148.T + 9.40e3T^{2} \) |
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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
−14.91955880840339159707893398406, −13.66645924388582348919696726975, −12.49455826704520582163906013315, −11.17502637610780068824005350788, −9.972752325570374667774493372978, −8.829296148460901554638423651323, −7.71073341825078955465255062435, −6.10997495613642719678911810278, −4.51759954463942933635290356333, −2.64128097419712183634720494822,
2.16489426123888751896543122692, 3.43334603817897659111892600415, 6.42754878807846510025756456070, 6.98151894173175420380022210400, 8.423300769362383091382898753903, 10.03017643702558325413081949463, 10.99455764337689915798389293961, 12.37919793050491912755560687617, 12.99644208755244073979439162501, 14.62859535876241314760972889818