L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 11-s − 12-s − 2·14-s + 16-s − 17-s + 18-s − 4·19-s + 2·21-s − 22-s + 6·23-s − 24-s − 5·25-s − 27-s − 2·28-s − 8·29-s − 2·31-s + 32-s + 33-s − 34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.436·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s − 25-s − 0.192·27-s − 0.377·28-s − 1.48·29-s − 0.359·31-s + 0.176·32-s + 0.174·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.45225054227407, −13.00741841549546, −12.64954402261580, −12.45687724157773, −11.59842793953549, −11.32674182851570, −10.85459810317176, −10.48779814225760, −9.817359969597186, −9.428018612744845, −8.946765393007951, −8.275785446823183, −7.559931864761269, −7.370295624582283, −6.538124404045714, −6.363156481467822, −5.849174442730252, −5.199342154645516, −4.881277759746007, −4.191202404488151, −3.705016945803747, −3.160036165161292, −2.591761198729176, −1.819318398478427, −1.351188048040379, 0, 0,
1.351188048040379, 1.819318398478427, 2.591761198729176, 3.160036165161292, 3.705016945803747, 4.191202404488151, 4.881277759746007, 5.199342154645516, 5.849174442730252, 6.363156481467822, 6.538124404045714, 7.370295624582283, 7.559931864761269, 8.275785446823183, 8.946765393007951, 9.428018612744845, 9.817359969597186, 10.48779814225760, 10.85459810317176, 11.32674182851570, 11.59842793953549, 12.45687724157773, 12.64954402261580, 13.00741841549546, 13.45225054227407