L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s − 4·11-s + 2·13-s − 16-s − 6·17-s − 4·19-s + 2·20-s − 4·22-s − 25-s + 2·26-s + 2·29-s + 5·32-s − 6·34-s + 6·37-s − 4·38-s + 6·40-s + 2·41-s − 4·43-s + 4·44-s − 50-s − 2·52-s − 6·53-s + 8·55-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s − 1.20·11-s + 0.554·13-s − 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.447·20-s − 0.852·22-s − 1/5·25-s + 0.392·26-s + 0.371·29-s + 0.883·32-s − 1.02·34-s + 0.986·37-s − 0.648·38-s + 0.948·40-s + 0.312·41-s − 0.609·43-s + 0.603·44-s − 0.141·50-s − 0.277·52-s − 0.824·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−10.92437047522813718231601421137, −9.800766905340478341584155035885, −8.609463409035164758983775505912, −8.123916451859490481483222873179, −6.81422856027217448872168795499, −5.73267711633132091527857634664, −4.61914309368534933978095799917, −3.94328917656394473634657471263, −2.64839746138688614832708523229, 0,
2.64839746138688614832708523229, 3.94328917656394473634657471263, 4.61914309368534933978095799917, 5.73267711633132091527857634664, 6.81422856027217448872168795499, 8.123916451859490481483222873179, 8.609463409035164758983775505912, 9.800766905340478341584155035885, 10.92437047522813718231601421137