| L(s) = 1 | + 2·3-s + 3·9-s + 4·19-s + 10·27-s − 40·43-s + 28·49-s + 8·57-s − 28·67-s + 4·73-s + 20·81-s + 40·97-s − 14·121-s + 127-s − 80·129-s + 131-s + 137-s + 139-s + 56·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 12·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 0.917·19-s + 1.92·27-s − 6.09·43-s + 4·49-s + 1.05·57-s − 3.42·67-s + 0.468·73-s + 20/9·81-s + 4.06·97-s − 1.27·121-s + 0.0887·127-s − 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.61·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.917·171-s + 0.0760·173-s + 0.0747·179-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
| \(L(1)\) |
\(\approx\) |
\(6.709823683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.709823683\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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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
−6.29810652928244661522904869651, −6.23583169142696017185866501008, −6.04309951234332684605495159415, −5.64228958804215514425690321768, −5.61648469931404268091350290471, −5.21243116302503239715992755814, −4.95728637145093159668528647275, −4.94986362810558856804720390121, −4.67956448623710692734432629639, −4.58871857086127085476298681174, −4.20759989122704855256178912501, −3.78296992102574489846456675371, −3.76051372594321453718201247905, −3.51420331642559724177430194377, −3.43319629024432813609016860739, −2.92631142086972117996289086165, −2.84611926641799227065955828054, −2.75574903041701673804387748428, −2.26856364027439428353085544349, −2.11014474745977619101197590496, −1.63666091268367869758796457189, −1.42563044213681355331281555974, −1.40171065968265319313764330264, −0.64392262530148553165785526353, −0.44265595158499176382231464661,
0.44265595158499176382231464661, 0.64392262530148553165785526353, 1.40171065968265319313764330264, 1.42563044213681355331281555974, 1.63666091268367869758796457189, 2.11014474745977619101197590496, 2.26856364027439428353085544349, 2.75574903041701673804387748428, 2.84611926641799227065955828054, 2.92631142086972117996289086165, 3.43319629024432813609016860739, 3.51420331642559724177430194377, 3.76051372594321453718201247905, 3.78296992102574489846456675371, 4.20759989122704855256178912501, 4.58871857086127085476298681174, 4.67956448623710692734432629639, 4.94986362810558856804720390121, 4.95728637145093159668528647275, 5.21243116302503239715992755814, 5.61648469931404268091350290471, 5.64228958804215514425690321768, 6.04309951234332684605495159415, 6.23583169142696017185866501008, 6.29810652928244661522904869651
