L(s) = 1 | + i·3-s + (1.34 − 1.78i)5-s − 0.521i·7-s − 9-s − 3.11·11-s + 5.31i·13-s + (1.78 + 1.34i)15-s − 5.37i·17-s + 0.478·19-s + 0.521·21-s + 9.04i·23-s + (−1.39 − 4.80i)25-s − i·27-s + 3.80·29-s + 3.77·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (0.600 − 0.799i)5-s − 0.197i·7-s − 0.333·9-s − 0.938·11-s + 1.47i·13-s + (0.461 + 0.346i)15-s − 1.30i·17-s + 0.109·19-s + 0.113·21-s + 1.88i·23-s + (−0.278 − 0.960i)25-s − 0.192i·27-s + 0.707·29-s + 0.678·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.811247796\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.811247796\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.34 + 1.78i)T \) |
| 67 | \( 1 - iT \) |
good | 7 | \( 1 + 0.521iT - 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 - 5.31iT - 13T^{2} \) |
| 17 | \( 1 + 5.37iT - 17T^{2} \) |
| 19 | \( 1 - 0.478T + 19T^{2} \) |
| 23 | \( 1 - 9.04iT - 23T^{2} \) |
| 29 | \( 1 - 3.80T + 29T^{2} \) |
| 31 | \( 1 - 3.77T + 31T^{2} \) |
| 37 | \( 1 + 5.87iT - 37T^{2} \) |
| 41 | \( 1 - 5.50T + 41T^{2} \) |
| 43 | \( 1 + 2.92iT - 43T^{2} \) |
| 47 | \( 1 - 11.7iT - 47T^{2} \) |
| 53 | \( 1 - 13.8iT - 53T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 + 7.47T + 61T^{2} \) |
| 71 | \( 1 + 4.15T + 71T^{2} \) |
| 73 | \( 1 - 8.31iT - 73T^{2} \) |
| 79 | \( 1 - 6.27T + 79T^{2} \) |
| 83 | \( 1 - 5.47iT - 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 18.8iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.859532322660821329560434066139, −7.74607414027992645510574276525, −7.25829785286431857107454062538, −6.11878273893715520746025258858, −5.52176545339373535880448105571, −4.70660977348337236032235167568, −4.26156189652818864544118729755, −3.03973002788348626122981743511, −2.16261011455400297016162657699, −0.986125597329853064808572789622,
0.60274293960759654977893877744, 2.01437506425617340313570322868, 2.71680646348798278940140222482, 3.39407138339606113237251018612, 4.72868434964370738134756240688, 5.54104573785958133167429419016, 6.23032014787069342914990236211, 6.69869991197752920954493281366, 7.76013856225658661879121600738, 8.161714310622497239750134638518