L(s) = 1 | + (0.552 − 0.319i)2-s + (−0.662 − 1.60i)3-s + (−0.796 + 1.37i)4-s + (−0.5 + 0.866i)5-s + (−0.877 − 0.672i)6-s + (−1.77 + 1.96i)7-s + 2.29i·8-s + (−2.12 + 2.12i)9-s + 0.638i·10-s + (−2.22 + 1.28i)11-s + (2.73 + 0.359i)12-s + (−2.91 − 1.68i)13-s + (−0.353 + 1.65i)14-s + (1.71 + 0.226i)15-s + (−0.860 − 1.49i)16-s + 2.60·17-s + ⋯ |
L(s) = 1 | + (0.390 − 0.225i)2-s + (−0.382 − 0.923i)3-s + (−0.398 + 0.689i)4-s + (−0.223 + 0.387i)5-s + (−0.358 − 0.274i)6-s + (−0.670 + 0.742i)7-s + 0.810i·8-s + (−0.707 + 0.707i)9-s + 0.201i·10-s + (−0.671 + 0.387i)11-s + (0.789 + 0.103i)12-s + (−0.807 − 0.466i)13-s + (−0.0944 + 0.441i)14-s + (0.443 + 0.0583i)15-s + (−0.215 − 0.372i)16-s + 0.632·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.395024 + 0.495989i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.395024 + 0.495989i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.662 + 1.60i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.77 - 1.96i)T \) |
good | 2 | \( 1 + (-0.552 + 0.319i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (2.22 - 1.28i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.91 + 1.68i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 - 3.95iT - 19T^{2} \) |
| 23 | \( 1 + (-4.63 - 2.67i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.59 - 1.50i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.64 + 3.26i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 + (2.74 - 4.74i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.78 + 8.29i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.53 - 9.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8.57iT - 53T^{2} \) |
| 59 | \( 1 + (4.28 - 7.41i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.81 - 2.20i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.92 - 8.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.71iT - 71T^{2} \) |
| 73 | \( 1 + 6.51iT - 73T^{2} \) |
| 79 | \( 1 + (1.07 + 1.85i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.62 - 6.28i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + (0.235 - 0.136i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.11913004177826669393472383231, −11.39829779762250281110502056461, −10.18130958469031948252098145563, −9.007691425870077730155399159470, −7.80801391745441087163590977422, −7.34255755574860349197348907995, −5.89617546714233810534360707129, −5.06864585748392931481300078307, −3.35484791058124487601919039622, −2.40167386431401754442630419995,
0.41008130297162975931925027995, 3.30055558728072535354649188208, 4.49331719339297258420012279805, 5.15407634969725837722037434845, 6.23879830840259877071597119488, 7.34092952767051773823351563879, 8.933841635946294264508561046337, 9.596964189901215347822182354242, 10.42996342846621856744821618254, 11.15215559237867745074495140968