L(s) = 1 | + 3-s − 3·7-s + 9-s − 4·13-s − 2·17-s + 2·19-s − 3·21-s − 7·23-s + 27-s − 6·29-s + 6·31-s − 10·37-s − 4·39-s − 2·41-s − 43-s + 7·47-s + 2·49-s − 2·51-s − 14·53-s + 2·57-s + 8·59-s + 2·61-s − 3·63-s + 11·67-s − 7·69-s − 4·71-s + 8·79-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 1.10·13-s − 0.485·17-s + 0.458·19-s − 0.654·21-s − 1.45·23-s + 0.192·27-s − 1.11·29-s + 1.07·31-s − 1.64·37-s − 0.640·39-s − 0.312·41-s − 0.152·43-s + 1.02·47-s + 2/7·49-s − 0.280·51-s − 1.92·53-s + 0.264·57-s + 1.04·59-s + 0.256·61-s − 0.377·63-s + 1.34·67-s − 0.842·69-s − 0.474·71-s + 0.900·79-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.35905112190858, −12.73818373085024, −12.42043618961100, −11.97768491048580, −11.51358686905823, −10.84190469476084, −10.25696271152997, −9.835080079191081, −9.669790263762161, −9.057819830920787, −8.593908691585694, −8.020960742725531, −7.534534011086625, −7.087204670313135, −6.573299917309893, −6.150203120064799, −5.499539006021491, −4.948195806154290, −4.397703979988527, −3.688468722785068, −3.414594384515565, −2.746216496674365, −2.164948350403565, −1.746353260130622, −0.6428148347703078, 0,
0.6428148347703078, 1.746353260130622, 2.164948350403565, 2.746216496674365, 3.414594384515565, 3.688468722785068, 4.397703979988527, 4.948195806154290, 5.499539006021491, 6.150203120064799, 6.573299917309893, 7.087204670313135, 7.534534011086625, 8.020960742725531, 8.593908691585694, 9.057819830920787, 9.669790263762161, 9.835080079191081, 10.25696271152997, 10.84190469476084, 11.51358686905823, 11.97768491048580, 12.42043618961100, 12.73818373085024, 13.35905112190858