L(s) = 1 | + 2-s + 2·3-s − 4-s + 2·5-s + 2·6-s + 2·7-s − 3·8-s + 9-s + 2·10-s − 6·11-s − 2·12-s − 13-s + 2·14-s + 4·15-s − 16-s + 17-s + 18-s + 4·19-s − 2·20-s + 4·21-s − 6·22-s + 6·23-s − 6·24-s − 25-s − 26-s − 4·27-s − 2·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.894·5-s + 0.816·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 1.80·11-s − 0.577·12-s − 0.277·13-s + 0.534·14-s + 1.03·15-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.872·21-s − 1.27·22-s + 1.25·23-s − 1.22·24-s − 1/5·25-s − 0.196·26-s − 0.769·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.176100076\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.176100076\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.81687262876098235036784582433, −11.41508377219727993221954013871, −10.10857183087426188490087859330, −9.297073189927582363453435681927, −8.359798380891594538656347298349, −7.48452778448443093986701738050, −5.57328576129429982536653103768, −5.03216113857164143727970023789, −3.37858390783506699471316087423, −2.31845179494526177837948227739,
2.31845179494526177837948227739, 3.37858390783506699471316087423, 5.03216113857164143727970023789, 5.57328576129429982536653103768, 7.48452778448443093986701738050, 8.359798380891594538656347298349, 9.297073189927582363453435681927, 10.10857183087426188490087859330, 11.41508377219727993221954013871, 12.81687262876098235036784582433