| L(s) = 1 | + 4-s − 2·7-s − 4·11-s − 3·16-s − 4·17-s + 4·19-s + 8·23-s − 2·28-s + 4·29-s + 12·31-s − 4·37-s + 4·41-s − 4·44-s + 8·47-s + 3·49-s − 16·53-s − 4·61-s − 7·64-s + 8·67-s − 4·68-s − 20·71-s + 16·73-s + 4·76-s + 8·77-s + 8·79-s − 16·83-s + 4·89-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 1.20·11-s − 3/4·16-s − 0.970·17-s + 0.917·19-s + 1.66·23-s − 0.377·28-s + 0.742·29-s + 2.15·31-s − 0.657·37-s + 0.624·41-s − 0.603·44-s + 1.16·47-s + 3/7·49-s − 2.19·53-s − 0.512·61-s − 7/8·64-s + 0.977·67-s − 0.485·68-s − 2.37·71-s + 1.87·73-s + 0.458·76-s + 0.911·77-s + 0.900·79-s − 1.75·83-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.707663101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707663101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.494345514010732904165717289736, −9.351316822599956564792789117632, −8.716652464724748411118915887606, −8.619683561142066412621774436704, −7.946633137850597036592330084622, −7.52075262924288314229924221466, −7.32355149476223724488835148879, −6.59366338742636364547223711507, −6.55397485941898522697951115704, −6.18065812543752856780268467809, −5.38929319082596267776795455533, −5.18030495844691240097084265071, −4.52919453867886346987795774115, −4.40076685660666986183249991522, −3.48193675663498596464661106456, −2.88483387594629201154961081634, −2.80139983874943462240685500440, −2.24356220133621027113733721003, −1.33560679219586659335090613942, −0.52187907817599172064592018163,
0.52187907817599172064592018163, 1.33560679219586659335090613942, 2.24356220133621027113733721003, 2.80139983874943462240685500440, 2.88483387594629201154961081634, 3.48193675663498596464661106456, 4.40076685660666986183249991522, 4.52919453867886346987795774115, 5.18030495844691240097084265071, 5.38929319082596267776795455533, 6.18065812543752856780268467809, 6.55397485941898522697951115704, 6.59366338742636364547223711507, 7.32355149476223724488835148879, 7.52075262924288314229924221466, 7.946633137850597036592330084622, 8.619683561142066412621774436704, 8.716652464724748411118915887606, 9.351316822599956564792789117632, 9.494345514010732904165717289736
