| L(s) = 1 | − 12·17-s − 20·25-s + 22·49-s − 32·61-s − 44·73-s − 72·101-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 2.91·17-s − 4·25-s + 22/7·49-s − 4.09·61-s − 5.14·73-s − 7.16·101-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2116373121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2116373121\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 127 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.17541702756754149040449313787, −6.15337043931015113580391178321, −5.92220026795083853302418081219, −5.68914233421032960251144747281, −5.38082830552768655783380532580, −5.37689798950933125439467544931, −5.16814193085337075546575211458, −4.48900353831397548406002460133, −4.41556998186001713596199870720, −4.41263932494147596938089858846, −4.31972377085614825407596437555, −4.04113835189103264948777563754, −3.82872867498499349140443222270, −3.55785496832310287662216128186, −3.26361250636276239036894271178, −2.81357402960029076092782429603, −2.75032827747300981486839392581, −2.52369903286551647148753804440, −2.30924569151904736462449015136, −1.90502802454233838702577280717, −1.73132061787377782261939179435, −1.38183004231919183301121952735, −1.34370502052434864237941298728, −0.38230660133963674759771497658, −0.12157138042056293530031649410,
0.12157138042056293530031649410, 0.38230660133963674759771497658, 1.34370502052434864237941298728, 1.38183004231919183301121952735, 1.73132061787377782261939179435, 1.90502802454233838702577280717, 2.30924569151904736462449015136, 2.52369903286551647148753804440, 2.75032827747300981486839392581, 2.81357402960029076092782429603, 3.26361250636276239036894271178, 3.55785496832310287662216128186, 3.82872867498499349140443222270, 4.04113835189103264948777563754, 4.31972377085614825407596437555, 4.41263932494147596938089858846, 4.41556998186001713596199870720, 4.48900353831397548406002460133, 5.16814193085337075546575211458, 5.37689798950933125439467544931, 5.38082830552768655783380532580, 5.68914233421032960251144747281, 5.92220026795083853302418081219, 6.15337043931015113580391178321, 6.17541702756754149040449313787
