| L(s) = 1 | + 4-s + 2·7-s − 10·13-s + 4·16-s − 4·19-s − 2·25-s + 2·28-s − 10·31-s − 4·37-s + 2·43-s + 15·49-s − 10·52-s − 4·61-s + 11·64-s − 16·67-s + 8·73-s − 4·76-s + 2·79-s − 20·91-s − 34·97-s − 2·100-s − 16·103-s + 68·109-s + 8·112-s + 10·121-s − 10·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s − 2.77·13-s + 16-s − 0.917·19-s − 2/5·25-s + 0.377·28-s − 1.79·31-s − 0.657·37-s + 0.304·43-s + 15/7·49-s − 1.38·52-s − 0.512·61-s + 11/8·64-s − 1.95·67-s + 0.936·73-s − 0.458·76-s + 0.225·79-s − 2.09·91-s − 3.45·97-s − 1/5·100-s − 1.57·103-s + 6.51·109-s + 0.755·112-s + 0.909·121-s − 0.898·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.473708930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.473708930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^3$ | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 2 T^{2} - 525 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 46 T^{2} + 1275 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 70 T^{2} + 3219 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 82 T^{2} + 4515 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 118 T^{2} + 7035 T^{4} - 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.789323586165361484461357426998, −8.608723514576209367941208707134, −8.241147477535537178114824731863, −8.006903235802678557635785973711, −7.82704421819917689819318639729, −7.39680556639963975985484355977, −7.18437291149639858644195074932, −7.05956508171912337153803989178, −7.02881533533033942315144179759, −6.51539851266405452569228860040, −5.93982797556827404509640005013, −5.76198737946871136007362888362, −5.64648580688680913343882915749, −5.32027457550671205414922003503, −4.96584975226237627380466797561, −4.52016100892549047316912626960, −4.37094827743046545745590114849, −4.20465437540909186227380024054, −3.49625084704881218547416639341, −3.15987605895427165078636724539, −2.94595583608623063452189652700, −2.17825215517418139029996194373, −2.03954259792561744116669193558, −1.83041881739976419156957473316, −0.62500319074072478064120732846,
0.62500319074072478064120732846, 1.83041881739976419156957473316, 2.03954259792561744116669193558, 2.17825215517418139029996194373, 2.94595583608623063452189652700, 3.15987605895427165078636724539, 3.49625084704881218547416639341, 4.20465437540909186227380024054, 4.37094827743046545745590114849, 4.52016100892549047316912626960, 4.96584975226237627380466797561, 5.32027457550671205414922003503, 5.64648580688680913343882915749, 5.76198737946871136007362888362, 5.93982797556827404509640005013, 6.51539851266405452569228860040, 7.02881533533033942315144179759, 7.05956508171912337153803989178, 7.18437291149639858644195074932, 7.39680556639963975985484355977, 7.82704421819917689819318639729, 8.006903235802678557635785973711, 8.241147477535537178114824731863, 8.608723514576209367941208707134, 8.789323586165361484461357426998
