L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s + 3·13-s − 17-s − 4·19-s + 2·21-s − 8·23-s − 27-s − 2·29-s − 8·31-s + 33-s − 8·37-s − 3·39-s − 6·41-s + 43-s + 47-s − 3·49-s + 51-s + 6·53-s + 4·57-s − 5·59-s + 4·61-s − 2·63-s − 11·67-s + 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.242·17-s − 0.917·19-s + 0.436·21-s − 1.66·23-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.174·33-s − 1.31·37-s − 0.480·39-s − 0.937·41-s + 0.152·43-s + 0.145·47-s − 3/7·49-s + 0.140·51-s + 0.824·53-s + 0.529·57-s − 0.650·59-s + 0.512·61-s − 0.251·63-s − 1.34·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.92438348726782, −14.65044369668991, −13.75579119671266, −13.41834757771046, −12.99870830356928, −12.38568837498927, −12.00113375590024, −11.41720161892009, −10.80499796683924, −10.36128946344396, −10.04457656434574, −9.244371269524564, −8.783554724890091, −8.269713111406903, −7.526784500828080, −7.015771512935515, −6.411969808639891, −5.919314396110145, −5.547720323045132, −4.738832814610570, −4.010230018829797, −3.658508808148833, −2.865486887556766, −1.960894598400226, −1.456533305989053, 0, 0,
1.456533305989053, 1.960894598400226, 2.865486887556766, 3.658508808148833, 4.010230018829797, 4.738832814610570, 5.547720323045132, 5.919314396110145, 6.411969808639891, 7.015771512935515, 7.526784500828080, 8.269713111406903, 8.783554724890091, 9.244371269524564, 10.04457656434574, 10.36128946344396, 10.80499796683924, 11.41720161892009, 12.00113375590024, 12.38568837498927, 12.99870830356928, 13.41834757771046, 13.75579119671266, 14.65044369668991, 14.92438348726782