L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s + 2·13-s + 14-s − 16-s + 3·17-s − 4·19-s − 6·23-s − 2·26-s + 28-s + 6·29-s + 9·31-s − 5·32-s − 3·34-s − 11·37-s + 4·38-s + 10·41-s − 9·43-s + 6·46-s − 47-s + 49-s − 2·52-s + 53-s − 3·56-s − 6·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.727·17-s − 0.917·19-s − 1.25·23-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 1.61·31-s − 0.883·32-s − 0.514·34-s − 1.80·37-s + 0.648·38-s + 1.56·41-s − 1.37·43-s + 0.884·46-s − 0.145·47-s + 1/7·49-s − 0.277·52-s + 0.137·53-s − 0.400·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7756599687\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7756599687\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.17259955632056, −12.60386431672873, −12.17718268396984, −11.72526647685671, −11.16444165795237, −10.38618925907679, −10.19836621706247, −10.02594113725966, −9.230157092569947, −8.824809328180022, −8.307906857686475, −8.095396793380780, −7.529604300254612, −6.702965794906272, −6.566463818913148, −5.773300385392540, −5.367265199751244, −4.630269427039625, −4.193014085000975, −3.733903272204056, −3.038628900588467, −2.402205947995264, −1.609771806753287, −1.101354051374805, −0.3207661673111128,
0.3207661673111128, 1.101354051374805, 1.609771806753287, 2.402205947995264, 3.038628900588467, 3.733903272204056, 4.193014085000975, 4.630269427039625, 5.367265199751244, 5.773300385392540, 6.566463818913148, 6.702965794906272, 7.529604300254612, 8.095396793380780, 8.307906857686475, 8.824809328180022, 9.230157092569947, 10.02594113725966, 10.19836621706247, 10.38618925907679, 11.16444165795237, 11.72526647685671, 12.17718268396984, 12.60386431672873, 13.17259955632056