L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s − 7-s + 8-s + 9-s − 2·10-s − 11-s + 12-s − 14-s − 2·15-s + 16-s − 17-s + 18-s + 4·19-s − 2·20-s − 21-s − 22-s − 6·23-s + 24-s − 25-s + 27-s − 28-s − 6·29-s − 2·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.365·30-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 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.35067275790480, −12.86295669317117, −12.53373852879626, −12.00345189669233, −11.50700024721743, −11.14572900364113, −10.69152871842050, −9.926270480773207, −9.583603184871406, −9.215294828892630, −8.352872253549553, −7.973710156635226, −7.684800499371684, −7.119859765965690, −6.638483886451748, −6.024478334448759, −5.445975940805274, −5.042225678995937, −4.212856716044062, −3.864991918671456, −3.557254352893917, −2.874077115315608, −2.289626055201310, −1.755594830835895, −0.8103544642618587, 0,
0.8103544642618587, 1.755594830835895, 2.289626055201310, 2.874077115315608, 3.557254352893917, 3.864991918671456, 4.212856716044062, 5.042225678995937, 5.445975940805274, 6.024478334448759, 6.638483886451748, 7.119859765965690, 7.684800499371684, 7.973710156635226, 8.352872253549553, 9.215294828892630, 9.583603184871406, 9.926270480773207, 10.69152871842050, 11.14572900364113, 11.50700024721743, 12.00345189669233, 12.53373852879626, 12.86295669317117, 13.35067275790480