| L(s) = 1 | − 9-s + 8·13-s + 6·17-s + 8·29-s + 16·37-s + 10·41-s − 14·49-s + 8·53-s + 16·61-s + 18·73-s − 8·81-s + 30·89-s + 4·97-s + 2·113-s − 8·117-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.21·13-s + 1.45·17-s + 1.48·29-s + 2.63·37-s + 1.56·41-s − 2·49-s + 1.09·53-s + 2.04·61-s + 2.10·73-s − 8/9·81-s + 3.17·89-s + 0.406·97-s + 0.188·113-s − 0.739·117-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.485·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.059062624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.059062624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 121 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.178833739113274228790155030005, −7.87578198971099780593245180647, −7.71200258701202988527179133311, −7.18457468321643443753253307762, −6.56506660099144910712706362383, −6.42148271699315207010383670336, −6.04193685492397176513380800139, −5.95686343036384286121563316093, −5.26939069807350142130492209008, −5.12741908486885863128051115047, −4.60021140137269249809631041711, −4.08644891454800828178811114779, −3.67659336810190408368152696911, −3.62924183486464120899544522075, −2.89342311514597227421484031791, −2.67241019895291349767595549830, −2.13295121587361647972285050240, −1.39064664265563924031973611622, −0.892112337888599768436419517508, −0.78778378874472832130220495508,
0.78778378874472832130220495508, 0.892112337888599768436419517508, 1.39064664265563924031973611622, 2.13295121587361647972285050240, 2.67241019895291349767595549830, 2.89342311514597227421484031791, 3.62924183486464120899544522075, 3.67659336810190408368152696911, 4.08644891454800828178811114779, 4.60021140137269249809631041711, 5.12741908486885863128051115047, 5.26939069807350142130492209008, 5.95686343036384286121563316093, 6.04193685492397176513380800139, 6.42148271699315207010383670336, 6.56506660099144910712706362383, 7.18457468321643443753253307762, 7.71200258701202988527179133311, 7.87578198971099780593245180647, 8.178833739113274228790155030005
