L(s) = 1 | + (0.866 + 0.5i)3-s + (−0.5 − 0.866i)5-s + i·7-s + (−0.866 − 0.5i)11-s − 0.999i·15-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.5 + 0.866i)21-s + (0.866 − 0.5i)23-s − i·27-s + (−0.866 − 0.5i)31-s + (−0.499 − 0.866i)33-s + (0.866 − 0.5i)35-s + (0.5 + 0.866i)37-s + (−0.866 + 0.5i)47-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)3-s + (−0.5 − 0.866i)5-s + i·7-s + (−0.866 − 0.5i)11-s − 0.999i·15-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.5 + 0.866i)21-s + (0.866 − 0.5i)23-s − i·27-s + (−0.866 − 0.5i)31-s + (−0.499 − 0.866i)33-s + (0.866 − 0.5i)35-s + (0.5 + 0.866i)37-s + (−0.866 + 0.5i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8124955692\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8124955692\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−12.66992477175824335103398277134, −11.63504597964918013150454296851, −10.50908870190490233697565250461, −9.309996863911135671874973673591, −8.499984918113276897785770765506, −8.152010468097277936804154069629, −6.27309012268999072264489363596, −5.04202855766530028312325826175, −3.84728006389649320724127329225, −2.50306402926853600546352933742,
2.37913706468566554218222193660, 3.51637984074493596937248463765, 4.97377363559341987547388628072, 6.96509250196731627601181733578, 7.30376574637874454210651227516, 8.294787861306813368008114044733, 9.495355798462048788247802205096, 10.77165902298516938071181958834, 11.18352389090318743678723269802, 12.80655480327646462813155737048