L(s) = 1 | + (−2.64 − 1.41i)3-s − 7.48i·5-s − 5.29·7-s + (5 + 7.48i)9-s + 14.1i·11-s − 10·13-s + (−10.5 + 19.7i)15-s − 26.4·19-s + (14.0 + 7.48i)21-s − 16.9i·23-s − 31·25-s + (−2.64 − 26.8i)27-s + 37.4i·29-s − 26.4·31-s + (20.0 − 37.4i)33-s + ⋯ |
L(s) = 1 | + (−0.881 − 0.471i)3-s − 1.49i·5-s − 0.755·7-s + (0.555 + 0.831i)9-s + 1.28i·11-s − 0.769·13-s + (−0.705 + 1.31i)15-s − 1.39·19-s + (0.666 + 0.356i)21-s − 0.737i·23-s − 1.23·25-s + (−0.0979 − 0.995i)27-s + 1.29i·29-s − 0.853·31-s + (0.606 − 1.13i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0440522 + 0.175863i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0440522 + 0.175863i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.64 + 1.41i)T \) |
good | 5 | \( 1 + 7.48iT - 25T^{2} \) |
| 7 | \( 1 + 5.29T + 49T^{2} \) |
| 11 | \( 1 - 14.1iT - 121T^{2} \) |
| 13 | \( 1 + 10T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 26.4T + 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 - 37.4iT - 841T^{2} \) |
| 31 | \( 1 + 26.4T + 961T^{2} \) |
| 37 | \( 1 + 10T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 58.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 37.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 98.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 90T + 3.72e3T^{2} \) |
| 67 | \( 1 + 5.29T + 4.48e3T^{2} \) |
| 71 | \( 1 + 28.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 30T + 5.32e3T^{2} \) |
| 79 | \( 1 + 26.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 25.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 74.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 10T + 9.40e3T^{2} \) |
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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
−12.25749887958143053903097787535, −10.76886171697768760479996516101, −9.762106987276690578847406369218, −8.783553357866111495385807092185, −7.48087529083086005919033918143, −6.47399744890267992836552336299, −5.17294950887508157696896726092, −4.40894961992028510431517457752, −1.88794707335394461039887543644, −0.11136600134054828665966240242,
2.86409261275417217904979543644, 4.03256220575157467098998901306, 5.81969621941266909414881572127, 6.42965121271951234050653037521, 7.49458689386566053176912035398, 9.181859514434677727198227347599, 10.21699412196717432550178196443, 10.85825559791518553308835389255, 11.61041272090993498315965704444, 12.75036662506769114573545260698