| L(s) = 1 | + 4·7-s − 9-s − 4·17-s + 8·23-s − 2·25-s + 12·31-s + 4·41-s − 8·47-s + 6·49-s − 4·63-s − 8·71-s + 4·73-s − 4·79-s + 81-s − 20·89-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 16·119-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1/3·9-s − 0.970·17-s + 1.66·23-s − 2/5·25-s + 2.15·31-s + 0.624·41-s − 1.16·47-s + 6/7·49-s − 0.503·63-s − 0.949·71-s + 0.468·73-s − 0.450·79-s + 1/9·81-s − 2.11·89-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.46·119-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.485325754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.485325754\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−15.1159642368, −14.5788986647, −14.1380051130, −13.8513064788, −13.1402666476, −12.9423291865, −12.1731538721, −11.5961956681, −11.3975698782, −10.9751603747, −10.4350974839, −9.85400535143, −9.21633470210, −8.66505084843, −8.30396687205, −7.82361264423, −7.15991074616, −6.60592317928, −5.96132634090, −5.18655382232, −4.70056166193, −4.26644918879, −3.12621586575, −2.37042974349, −1.29102808548,
1.29102808548, 2.37042974349, 3.12621586575, 4.26644918879, 4.70056166193, 5.18655382232, 5.96132634090, 6.60592317928, 7.15991074616, 7.82361264423, 8.30396687205, 8.66505084843, 9.21633470210, 9.85400535143, 10.4350974839, 10.9751603747, 11.3975698782, 11.5961956681, 12.1731538721, 12.9423291865, 13.1402666476, 13.8513064788, 14.1380051130, 14.5788986647, 15.1159642368
