L(s) = 1 | + (−0.707 + 0.707i)5-s − 4.82i·7-s + (1.41 − 1.41i)11-s + (−0.585 − 0.585i)13-s − 5.41·17-s + (−3.82 − 3.82i)19-s + 5.41i·23-s − 1.00i·25-s + (0.585 + 0.585i)29-s − 3.65·31-s + (3.41 + 3.41i)35-s + (4.58 − 4.58i)37-s + 4.82i·41-s + (−3.65 + 3.65i)43-s − 7.07·47-s + ⋯ |
L(s) = 1 | + (−0.316 + 0.316i)5-s − 1.82i·7-s + (0.426 − 0.426i)11-s + (−0.162 − 0.162i)13-s − 1.31·17-s + (−0.878 − 0.878i)19-s + 1.12i·23-s − 0.200i·25-s + (0.108 + 0.108i)29-s − 0.656·31-s + (0.577 + 0.577i)35-s + (0.753 − 0.753i)37-s + 0.754i·41-s + (−0.557 + 0.557i)43-s − 1.03·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2411612824\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2411612824\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + 4.82iT - 7T^{2} \) |
| 11 | \( 1 + (-1.41 + 1.41i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.585 + 0.585i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 + (3.82 + 3.82i)T + 19iT^{2} \) |
| 23 | \( 1 - 5.41iT - 23T^{2} \) |
| 29 | \( 1 + (-0.585 - 0.585i)T + 29iT^{2} \) |
| 31 | \( 1 + 3.65T + 31T^{2} \) |
| 37 | \( 1 + (-4.58 + 4.58i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.82iT - 41T^{2} \) |
| 43 | \( 1 + (3.65 - 3.65i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 + (-4 + 4i)T - 53iT^{2} \) |
| 59 | \( 1 + (7.41 - 7.41i)T - 59iT^{2} \) |
| 61 | \( 1 + (-9.48 - 9.48i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.65 - 7.65i)T + 67iT^{2} \) |
| 71 | \( 1 - 8iT - 71T^{2} \) |
| 73 | \( 1 + 3.17iT - 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + (3.07 + 3.07i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.65iT - 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.255272445699174556812320987213, −7.47885034294886632250932888592, −6.87638957963966682733597490316, −6.35916235041932959127136366652, −5.08799199223083203900548798445, −4.17484155483564348684506678060, −3.77387298925688915157723342194, −2.62920416417805850310614101208, −1.26912388029076335043821907003, −0.07657363644073475144080116209,
1.88475167496763988928312597648, 2.42924184599820700324495581500, 3.68255788747597749968885381143, 4.63084691622531435835797282866, 5.25881038360730769682031326321, 6.33028208385152991339262919303, 6.62596797254418768842114041720, 7.977705340069364437796660401183, 8.492254692077510987816339139981, 9.092476159867671185292337413089