L(s) = 1 | + 3-s − 5·7-s − 2·9-s + 5·11-s − 13-s − 4·17-s + 4·19-s − 5·21-s − 9·23-s − 5·25-s − 5·27-s − 2·29-s + 2·31-s + 5·33-s + 6·37-s − 39-s + 6·41-s + 5·43-s + 47-s + 18·49-s − 4·51-s + 6·53-s + 4·57-s + 12·59-s + 3·61-s + 10·63-s + 5·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.88·7-s − 2/3·9-s + 1.50·11-s − 0.277·13-s − 0.970·17-s + 0.917·19-s − 1.09·21-s − 1.87·23-s − 25-s − 0.962·27-s − 0.371·29-s + 0.359·31-s + 0.870·33-s + 0.986·37-s − 0.160·39-s + 0.937·41-s + 0.762·43-s + 0.145·47-s + 18/7·49-s − 0.560·51-s + 0.824·53-s + 0.529·57-s + 1.56·59-s + 0.384·61-s + 1.25·63-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.47376975740131, −14.59493339892363, −14.25640911284373, −13.76946290262108, −13.28430555159585, −12.76611230479285, −12.09324085260408, −11.66191484922681, −11.27099531557229, −10.21609362465866, −9.795196386031882, −9.419108989369483, −8.983751027250435, −8.406973272546035, −7.611590826776944, −7.114751561490427, −6.353384013616005, −6.064231891748476, −5.556539695280312, −4.288982848453137, −3.838447025093696, −3.466520095730209, −2.514703699173238, −2.200257605456542, −0.8829838094186476, 0,
0.8829838094186476, 2.200257605456542, 2.514703699173238, 3.466520095730209, 3.838447025093696, 4.288982848453137, 5.556539695280312, 6.064231891748476, 6.353384013616005, 7.114751561490427, 7.611590826776944, 8.406973272546035, 8.983751027250435, 9.419108989369483, 9.795196386031882, 10.21609362465866, 11.27099531557229, 11.66191484922681, 12.09324085260408, 12.76611230479285, 13.28430555159585, 13.76946290262108, 14.25640911284373, 14.59493339892363, 15.47376975740131