| L(s) = 1 | + 2·3-s + 2·5-s + 4·7-s − 2·13-s + 4·15-s + 2·19-s + 8·21-s + 3·25-s − 2·27-s + 4·31-s + 8·35-s + 10·37-s − 4·39-s − 12·41-s + 4·43-s + 12·47-s − 2·49-s − 18·53-s + 4·57-s + 4·61-s − 4·65-s + 10·67-s − 12·71-s − 8·73-s + 6·75-s − 8·79-s − 81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 1.51·7-s − 0.554·13-s + 1.03·15-s + 0.458·19-s + 1.74·21-s + 3/5·25-s − 0.384·27-s + 0.718·31-s + 1.35·35-s + 1.64·37-s − 0.640·39-s − 1.87·41-s + 0.609·43-s + 1.75·47-s − 2/7·49-s − 2.47·53-s + 0.529·57-s + 0.512·61-s − 0.496·65-s + 1.22·67-s − 1.42·71-s − 0.936·73-s + 0.692·75-s − 0.900·79-s − 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.224583453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.224583453\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 96 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 156 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 310 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T - 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.44331589690108080271688966417, −11.24458296164480425319556749824, −10.59458783679243013300771317226, −10.16504772441996179259738013179, −9.483940699053012872537189687096, −9.477293355440750799891256755494, −8.665256492777781791062861870619, −8.425170139690295072472280641962, −7.933651046001422676992367948173, −7.63467799182691072362611218413, −6.91665718516314033151540346617, −6.40148018371616747176468977557, −5.60739079263632553289990038191, −5.34029417693935189242495196863, −4.56237902716627680960834560191, −4.26524606504058423658859734907, −3.07160637666307823419567404448, −2.81262026871803154044574947246, −2.00908948934280587162148736490, −1.36351097618979905690069751548,
1.36351097618979905690069751548, 2.00908948934280587162148736490, 2.81262026871803154044574947246, 3.07160637666307823419567404448, 4.26524606504058423658859734907, 4.56237902716627680960834560191, 5.34029417693935189242495196863, 5.60739079263632553289990038191, 6.40148018371616747176468977557, 6.91665718516314033151540346617, 7.63467799182691072362611218413, 7.933651046001422676992367948173, 8.425170139690295072472280641962, 8.665256492777781791062861870619, 9.477293355440750799891256755494, 9.483940699053012872537189687096, 10.16504772441996179259738013179, 10.59458783679243013300771317226, 11.24458296164480425319556749824, 11.44331589690108080271688966417
