L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s + 2·13-s + 14-s + 16-s + 2·17-s + 4·19-s − 20-s − 22-s − 4·23-s + 25-s − 2·26-s − 28-s − 6·29-s − 32-s − 2·34-s + 35-s − 6·37-s − 4·38-s + 40-s − 10·41-s + 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.223·20-s − 0.213·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.176·32-s − 0.342·34-s + 0.169·35-s − 0.986·37-s − 0.648·38-s + 0.158·40-s − 1.56·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.55668647798830419085357635186, −7.15988469224783471362003142270, −6.26157602738731498974744104965, −5.68870223299217772115729983097, −4.77507433853185411557303488022, −3.63690558803954774689750301145, −3.33586462864031748207441515201, −2.09369528733945246161980100245, −1.17284818124080329186669045434, 0,
1.17284818124080329186669045434, 2.09369528733945246161980100245, 3.33586462864031748207441515201, 3.63690558803954774689750301145, 4.77507433853185411557303488022, 5.68870223299217772115729983097, 6.26157602738731498974744104965, 7.15988469224783471362003142270, 7.55668647798830419085357635186