| L(s) = 1 | + 20·13-s − 48·25-s + 48·37-s − 98·49-s − 240·61-s − 192·73-s + 288·97-s + 364·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.53·13-s − 1.91·25-s + 1.29·37-s − 2·49-s − 3.93·61-s − 2.63·73-s + 2.96·97-s + 3.33·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.224·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.632861072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632861072\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 48 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 480 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 1680 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 1440 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 5040 T^{2} + p^{4} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 120 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12480 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )^{2} \) |
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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
−8.842145459410476616902552962270, −8.729506866388548419513034495956, −8.317677849811205222985587324578, −7.73243599080167161080860550135, −7.50291655023378593815171580219, −7.37817366572728934989721998851, −6.36467510820816605927555506921, −6.31330561229614372636603538772, −5.91297342688333383548937838697, −5.78530157014760260693076419336, −4.88645056430772646822996890807, −4.61066120132458221895651249279, −4.26096575286916771012287081211, −3.57999328970603483483170588046, −3.36113294120507038391251910981, −2.88051548759532199293647657258, −2.09815366465067797241463584297, −1.65189854688914536201440344880, −1.18433847555581904953768810937, −0.32189122419786248184537346083,
0.32189122419786248184537346083, 1.18433847555581904953768810937, 1.65189854688914536201440344880, 2.09815366465067797241463584297, 2.88051548759532199293647657258, 3.36113294120507038391251910981, 3.57999328970603483483170588046, 4.26096575286916771012287081211, 4.61066120132458221895651249279, 4.88645056430772646822996890807, 5.78530157014760260693076419336, 5.91297342688333383548937838697, 6.31330561229614372636603538772, 6.36467510820816605927555506921, 7.37817366572728934989721998851, 7.50291655023378593815171580219, 7.73243599080167161080860550135, 8.317677849811205222985587324578, 8.729506866388548419513034495956, 8.842145459410476616902552962270
