L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·8-s − 0.999·10-s + (−0.5 + 0.866i)16-s − 2·19-s + (0.499 − 0.866i)20-s + (1 + 1.73i)23-s + (−0.499 + 0.866i)25-s + (−0.499 − 0.866i)32-s + (1 − 1.73i)38-s + (0.499 + 0.866i)40-s − 1.99·46-s + (−1 + 1.73i)47-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·8-s − 0.999·10-s + (−0.5 + 0.866i)16-s − 2·19-s + (0.499 − 0.866i)20-s + (1 + 1.73i)23-s + (−0.499 + 0.866i)25-s + (−0.499 − 0.866i)32-s + (1 − 1.73i)38-s + (0.499 + 0.866i)40-s − 1.99·46-s + (−1 + 1.73i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8462819566\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8462819566\) |
\(L(1)\) |
|
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 | \( 1 + (-0.5 - 0.866i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 2T + T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - 2T + T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.126653581760277880401808758774, −8.336888969505069459065654362275, −7.50100168289508229358595974334, −6.93266629202465018730620839480, −6.20679650971519158008739867262, −5.64562351207714329062667331361, −4.69389913923892919377927679603, −3.73124604204218145545801301187, −2.52820089214732696689282056988, −1.48004265556516971857384498110,
0.61504744121511205893253176862, 1.91957945246847552304129766681, 2.55805183169718558237202479785, 3.85415568563204848984734140095, 4.53804745236463145892881354344, 5.25819103182268195158043059908, 6.39125752813791901934506720462, 7.09814478025472966020028642326, 8.432555844609891135087778009843, 8.483802143388496471925786861488