| L(s) = 1 | + 4·7-s − 8·13-s − 16·17-s − 8·19-s + 8·37-s + 8·41-s − 8·43-s + 8·47-s + 8·49-s − 24·61-s − 8·67-s + 4·73-s + 32·79-s − 81-s + 16·83-s − 32·91-s + 20·97-s − 16·101-s + 36·103-s − 32·107-s + 8·113-s − 64·119-s + 8·121-s + 127-s + 131-s − 32·133-s + 137-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 2.21·13-s − 3.88·17-s − 1.83·19-s + 1.31·37-s + 1.24·41-s − 1.21·43-s + 1.16·47-s + 8/7·49-s − 3.07·61-s − 0.977·67-s + 0.468·73-s + 3.60·79-s − 1/9·81-s + 1.75·83-s − 3.35·91-s + 2.03·97-s − 1.59·101-s + 3.54·103-s − 3.09·107-s + 0.752·113-s − 5.86·119-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1915148022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1915148022\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 130 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 136 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 720 T^{3} + 3266 T^{4} + 720 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 958 T^{4} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 1106 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 70 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 328 T^{3} + 3346 T^{4} - 328 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 280 T^{3} + 2386 T^{4} + 280 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 152 T^{3} - 62 T^{4} - 152 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 88 T^{3} - 2894 T^{4} + 88 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4006 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 142 T^{2} + p^{2} T^{4} ) \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 240 T^{3} - 4174 T^{4} - 240 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 284 T^{2} + 35494 T^{4} - 284 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1660 T^{3} + 13582 T^{4} - 1660 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.43579818594736679861195358163, −6.12147507566435905506735015508, −5.96599219312285321110006211713, −5.79941759937005395390675490361, −5.62869714717252228599906954133, −5.06328149108773196608009831726, −4.99775713849581693732884211541, −4.86144077952031023979948289553, −4.53191441310544172214524668206, −4.48780627931657073593458327908, −4.47225781629570260009689131146, −4.20318078767036847158730727336, −4.03255175560117579180779769597, −3.52243527148268578992203019057, −3.21234486690165183818123038711, −3.13830043421826860682369391268, −2.59910319801823439419540182685, −2.38921155781413817955432560766, −2.15334108292271096658520023110, −2.06152842330464776909835296712, −2.04858440317627019538293018806, −1.65929121079174842021501631259, −0.969927713362424117892105184457, −0.67746247608905412412397876409, −0.083319365907436788962944679651,
0.083319365907436788962944679651, 0.67746247608905412412397876409, 0.969927713362424117892105184457, 1.65929121079174842021501631259, 2.04858440317627019538293018806, 2.06152842330464776909835296712, 2.15334108292271096658520023110, 2.38921155781413817955432560766, 2.59910319801823439419540182685, 3.13830043421826860682369391268, 3.21234486690165183818123038711, 3.52243527148268578992203019057, 4.03255175560117579180779769597, 4.20318078767036847158730727336, 4.47225781629570260009689131146, 4.48780627931657073593458327908, 4.53191441310544172214524668206, 4.86144077952031023979948289553, 4.99775713849581693732884211541, 5.06328149108773196608009831726, 5.62869714717252228599906954133, 5.79941759937005395390675490361, 5.96599219312285321110006211713, 6.12147507566435905506735015508, 6.43579818594736679861195358163
