L(s) = 1 | − 3-s + 2·7-s + 9-s + 2·13-s + 6·17-s + 8·19-s − 2·21-s + 6·23-s − 27-s + 6·29-s + 4·31-s + 8·37-s − 2·39-s + 6·41-s − 43-s + 6·47-s − 3·49-s − 6·51-s − 12·53-s − 8·57-s + 4·61-s + 2·63-s + 4·67-s − 6·69-s + 10·73-s − 8·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s + 1.83·19-s − 0.436·21-s + 1.25·23-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 1.31·37-s − 0.320·39-s + 0.937·41-s − 0.152·43-s + 0.875·47-s − 3/7·49-s − 0.840·51-s − 1.64·53-s − 1.05·57-s + 0.512·61-s + 0.251·63-s + 0.488·67-s − 0.722·69-s + 1.17·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.196455830\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.196455830\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−12.97086364333575, −12.39522858866560, −12.16627976531893, −11.41645670967934, −11.33118547798834, −10.86808276043651, −10.11072738794856, −9.893332251374029, −9.307128991705665, −8.848928065952381, −8.021872295322887, −7.869510772999998, −7.416250589783740, −6.694376992937288, −6.319224419612201, −5.566953042545452, −5.355386340518379, −4.791982947040114, −4.319740265216919, −3.574008136001184, −3.030336945786868, −2.584561005053478, −1.505679859233277, −1.062960283934296, −0.7536213597842804,
0.7536213597842804, 1.062960283934296, 1.505679859233277, 2.584561005053478, 3.030336945786868, 3.574008136001184, 4.319740265216919, 4.791982947040114, 5.355386340518379, 5.566953042545452, 6.319224419612201, 6.694376992937288, 7.416250589783740, 7.869510772999998, 8.021872295322887, 8.848928065952381, 9.307128991705665, 9.893332251374029, 10.11072738794856, 10.86808276043651, 11.33118547798834, 11.41645670967934, 12.16627976531893, 12.39522858866560, 12.97086364333575