L(s) = 1 | − 3·9-s − 4·13-s + 2i·17-s − 4i·29-s + 12·37-s + 10·41-s + 7·49-s + 4·53-s − 12i·61-s + 6i·73-s + 9·81-s + 10·89-s + 18i·97-s − 20i·101-s − 20i·109-s + ⋯ |
L(s) = 1 | − 9-s − 1.10·13-s + 0.485i·17-s − 0.742i·29-s + 1.97·37-s + 1.56·41-s + 49-s + 0.549·53-s − 1.53i·61-s + 0.702i·73-s + 81-s + 1.05·89-s + 1.82i·97-s − 1.99i·101-s − 1.91i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.394487793\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.394487793\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 12T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 12iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−8.566839174917558726316243266225, −7.896149526479872609028048004255, −7.25438679845608531451070043837, −6.21341210768247059980174048170, −5.70510013203006760095319576936, −4.76958668077760820362136380922, −3.97484686586189709183783656791, −2.84565555392979675698912581105, −2.20606770134580424980998439924, −0.61710629357465666210310724734,
0.76679370213445048722903756590, 2.35382097286369136464435644832, 2.89680729236596222818665924399, 4.06320426163702879380594545379, 4.92256337003355570242357659489, 5.65653825307123333157078792124, 6.38193377788816080786625430517, 7.39303259468716917458983305383, 7.80852877735676114772836925711, 8.864229234684621713217378287232