| L(s) = 1 | + 4-s + 3·5-s + 4·11-s + 16-s − 16·19-s + 3·20-s + 5·25-s + 12·29-s + 2·31-s − 12·41-s + 4·44-s + 4·49-s + 12·55-s − 18·59-s + 20·61-s + 5·64-s + 6·71-s − 16·76-s − 28·79-s + 3·80-s − 6·89-s − 48·95-s + 5·100-s − 24·101-s − 40·109-s + 12·116-s + 10·121-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.34·5-s + 1.20·11-s + 1/4·16-s − 3.67·19-s + 0.670·20-s + 25-s + 2.22·29-s + 0.359·31-s − 1.87·41-s + 0.603·44-s + 4/7·49-s + 1.61·55-s − 2.34·59-s + 2.56·61-s + 5/8·64-s + 0.712·71-s − 1.83·76-s − 3.15·79-s + 0.335·80-s − 0.635·89-s − 4.92·95-s + 1/2·100-s − 2.38·101-s − 3.83·109-s + 1.11·116-s + 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.215454424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.215454424\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T^{2} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 2856 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 6006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 9 T + 130 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 181 T^{2} + 15312 T^{4} - 181 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 337 T^{2} + 47136 T^{4} - 337 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.072589606435863488958085723105, −7.902745947851826464526717956306, −7.13492500598208375112746877626, −7.05896724375623888430722976920, −6.78315654329232781893009962934, −6.61905496234317664606825093475, −6.52607311725240580313014706959, −6.30107370512600189645130512392, −6.14104214213047503860029715752, −5.56954676142076890851381216933, −5.51715903748138664064835023440, −5.34215076587971729201867797199, −4.86852398142647905551644275263, −4.42398687116432921724375282634, −4.26373898850330753881250562788, −4.24239005125599039876232559681, −3.85187704678705133936033511036, −3.36464095299881239644023867390, −2.85763911746589908560085788073, −2.72587077208146909082375146012, −2.48389768317168587724938882949, −1.85078641146134540408760581349, −1.71972748890773063934424888471, −1.43741902355879981037806524267, −0.56609575285412674955697916984,
0.56609575285412674955697916984, 1.43741902355879981037806524267, 1.71972748890773063934424888471, 1.85078641146134540408760581349, 2.48389768317168587724938882949, 2.72587077208146909082375146012, 2.85763911746589908560085788073, 3.36464095299881239644023867390, 3.85187704678705133936033511036, 4.24239005125599039876232559681, 4.26373898850330753881250562788, 4.42398687116432921724375282634, 4.86852398142647905551644275263, 5.34215076587971729201867797199, 5.51715903748138664064835023440, 5.56954676142076890851381216933, 6.14104214213047503860029715752, 6.30107370512600189645130512392, 6.52607311725240580313014706959, 6.61905496234317664606825093475, 6.78315654329232781893009962934, 7.05896724375623888430722976920, 7.13492500598208375112746877626, 7.902745947851826464526717956306, 8.072589606435863488958085723105
