L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 0.999·8-s − 0.999·10-s + (1.5 − 0.866i)11-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (−0.499 + 0.866i)20-s − 1.73i·22-s + (−0.499 + 0.866i)25-s + (−0.499 + 0.866i)28-s − 1.73i·29-s + (−1.5 + 0.866i)31-s + (0.499 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s − 0.999·8-s − 0.999·10-s + (1.5 − 0.866i)11-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (−0.499 + 0.866i)20-s − 1.73i·22-s + (−0.499 + 0.866i)25-s + (−0.499 + 0.866i)28-s − 1.73i·29-s + (−1.5 + 0.866i)31-s + (0.499 + 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.144749545\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.144749545\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 11 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + 1.73iT - T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 1.73iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.965508751264370583553001526017, −8.130128431412474892657527125196, −7.05838674303407033731977007257, −6.21556891626706147134929833114, −5.45546657862314507995380533793, −4.33134954095277803176010502116, −3.92398034438817647942073644267, −3.18199777579852215653477321981, −1.63705852018038381288313948736, −0.66514774009648477496476025608,
2.11293767394826480970951434068, 3.30195290935388960386976135941, 3.83344984490323867162525628020, 4.81851983956314580832672086539, 5.79531457590504601716311614226, 6.52205480907579212904026471393, 7.03958467929972906683605482983, 7.68695170383956468189541907189, 8.788286625113986786108192580596, 9.194617174980421048290826583437