L(s) = 1 | + (−1.60 + 2.78i)5-s + (1.82 − 1.05i)7-s + (3.47 − 2.00i)11-s + (0.341 + 0.197i)13-s + 1.20i·17-s + 1.62·19-s + (−2.74 + 4.75i)23-s + (−2.68 − 4.64i)25-s + (2.95 + 5.12i)29-s + (3.34 + 1.93i)31-s + 6.77i·35-s + 10.8i·37-s + (1.23 + 0.715i)41-s + (1.21 + 2.10i)43-s + (−0.792 − 1.37i)47-s + ⋯ |
L(s) = 1 | + (−0.719 + 1.24i)5-s + (0.688 − 0.397i)7-s + (1.04 − 0.605i)11-s + (0.0948 + 0.0547i)13-s + 0.292i·17-s + 0.372·19-s + (−0.572 + 0.990i)23-s + (−0.536 − 0.928i)25-s + (0.549 + 0.950i)29-s + (0.601 + 0.347i)31-s + 1.14i·35-s + 1.77i·37-s + (0.193 + 0.111i)41-s + (0.185 + 0.321i)43-s + (−0.115 − 0.200i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28944 + 0.768272i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28944 + 0.768272i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.60 - 2.78i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.82 + 1.05i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.47 + 2.00i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.341 - 0.197i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.20iT - 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 + (2.74 - 4.75i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.95 - 5.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.34 - 1.93i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 + (-1.23 - 0.715i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.21 - 2.10i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.792 + 1.37i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 + (2.29 + 1.32i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.18 + 4.72i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.60 + 4.51i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 - 9.49T + 73T^{2} \) |
| 79 | \( 1 + (-1.53 + 0.886i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.30 - 0.755i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 11.2iT - 89T^{2} \) |
| 97 | \( 1 + (-5.84 - 10.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.45133068243404949480774285310, −9.575838018419266545647065235883, −8.410614937207613078041702969666, −7.77763644534487773266503975419, −6.85360553383949491999231687585, −6.23545964607285063534694744038, −4.86307032821467869251892017887, −3.76917398519598632098849105085, −3.09156102043624856686917532823, −1.40075515597191603813504251654,
0.844513951492844499489900901994, 2.19859876916455307839529385027, 3.95834411340823807088668536044, 4.54015230602621336393050520765, 5.44923805784093786858081477430, 6.58010287367600598573722965124, 7.71374064090926675545779287406, 8.350032633891855486283862745025, 9.073510614178348788642819104692, 9.813428510273030435929277677247