| L(s) = 1 | + 4·5-s + 9-s − 8·11-s + 4·19-s + 5·25-s − 24·29-s + 12·31-s + 4·45-s − 32·55-s − 8·59-s − 4·61-s − 32·71-s + 32·79-s − 32·89-s + 16·95-s − 8·99-s + 16·101-s + 36·109-s + 38·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 96·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1/3·9-s − 2.41·11-s + 0.917·19-s + 25-s − 4.45·29-s + 2.15·31-s + 0.596·45-s − 4.31·55-s − 1.04·59-s − 0.512·61-s − 3.79·71-s + 3.60·79-s − 3.39·89-s + 1.64·95-s − 0.804·99-s + 1.59·101-s + 3.44·109-s + 3.45·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 7.97·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1606985706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1606985706\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( ( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 78 T^{2} + 3875 T^{4} + 78 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 102 T^{2} + 7595 T^{4} + 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 50 T^{2} - 2829 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.05126431839718000420040690428, −5.92072147675580421453070366489, −5.89654474489274338284294546164, −5.67412978390412511833714221993, −5.43335278125285412818325819804, −5.14954077335541860969919494177, −5.08335084409772773506573169675, −4.88639726427674749393869043368, −4.57922510915999335267638290724, −4.46120449823935171556751709637, −4.27815002342273676576949134078, −3.69373856823323570745195890226, −3.64181961349167911041177023764, −3.46186318704185183757578909696, −3.21025064371054842045046101526, −2.87665743589810890999832571309, −2.57686774402473848559739337691, −2.40816305410420984106293619747, −2.39793839731160110273187275550, −1.83140658517299868192466036553, −1.74168765203007639876791434728, −1.56588118527570995651870460858, −1.14151916995512427173884168511, −0.68110247243081575114457671525, −0.06025621402253191250400830490,
0.06025621402253191250400830490, 0.68110247243081575114457671525, 1.14151916995512427173884168511, 1.56588118527570995651870460858, 1.74168765203007639876791434728, 1.83140658517299868192466036553, 2.39793839731160110273187275550, 2.40816305410420984106293619747, 2.57686774402473848559739337691, 2.87665743589810890999832571309, 3.21025064371054842045046101526, 3.46186318704185183757578909696, 3.64181961349167911041177023764, 3.69373856823323570745195890226, 4.27815002342273676576949134078, 4.46120449823935171556751709637, 4.57922510915999335267638290724, 4.88639726427674749393869043368, 5.08335084409772773506573169675, 5.14954077335541860969919494177, 5.43335278125285412818325819804, 5.67412978390412511833714221993, 5.89654474489274338284294546164, 5.92072147675580421453070366489, 6.05126431839718000420040690428
