L(s) = 1 | − 81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
L(s) = 1 | − 81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.101406186\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.101406186\) |
\(L(1)\) |
|
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 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.60154330534272279093343668521, −6.15258798723087000736068013009, −6.05318746570498503114681116315, −5.96399062176313524658500968717, −5.94233026105446575305931767642, −5.61195730519488714905864088034, −5.18001707253011900970451449656, −5.13959122435953503910722904017, −4.85542105385135321674884481014, −4.68960534326432404976060852232, −4.54893905190806380733199359028, −4.15376682563979416024946002433, −4.15350253278408281764083778630, −3.58237198836779789491750069634, −3.50511097719140906810486465285, −3.43716643650274088073445852885, −3.21863842390714793085334582859, −2.55292450455394578900997986732, −2.52685950090802457999639509183, −2.45977424059046015242562918504, −2.07378983996711400094650400249, −1.60592898770165627623366021782, −1.34262124958448198976749591894, −1.17123296346822384525524528466, −0.49538494907228287729740016498,
0.49538494907228287729740016498, 1.17123296346822384525524528466, 1.34262124958448198976749591894, 1.60592898770165627623366021782, 2.07378983996711400094650400249, 2.45977424059046015242562918504, 2.52685950090802457999639509183, 2.55292450455394578900997986732, 3.21863842390714793085334582859, 3.43716643650274088073445852885, 3.50511097719140906810486465285, 3.58237198836779789491750069634, 4.15350253278408281764083778630, 4.15376682563979416024946002433, 4.54893905190806380733199359028, 4.68960534326432404976060852232, 4.85542105385135321674884481014, 5.13959122435953503910722904017, 5.18001707253011900970451449656, 5.61195730519488714905864088034, 5.94233026105446575305931767642, 5.96399062176313524658500968717, 6.05318746570498503114681116315, 6.15258798723087000736068013009, 6.60154330534272279093343668521