| L(s) = 1 | + 8·7-s + 2·9-s + 4·13-s − 16-s − 12·19-s − 28·31-s − 4·37-s + 28·43-s + 12·49-s − 36·61-s + 16·63-s + 12·79-s − 5·81-s + 32·91-s + 28·97-s − 20·103-s − 12·109-s − 8·112-s + 8·117-s − 44·121-s + 127-s + 131-s − 96·133-s + 137-s + 139-s − 2·144-s + 149-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 2/3·9-s + 1.10·13-s − 1/4·16-s − 2.75·19-s − 5.02·31-s − 0.657·37-s + 4.26·43-s + 12/7·49-s − 4.60·61-s + 2.01·63-s + 1.35·79-s − 5/9·81-s + 3.35·91-s + 2.84·97-s − 1.97·103-s − 1.14·109-s − 0.755·112-s + 0.739·117-s − 4·121-s + 0.0887·127-s + 0.0873·131-s − 8.32·133-s + 0.0854·137-s + 0.0848·139-s − 1/6·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7133169852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7133169852\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 4046 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 14434 T^{4} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.14135111731970326643039348502, −6.86657709383484072232327963276, −6.58754683602131486024868992050, −6.20775614029697427791815305532, −6.19995369563461895392941089814, −6.02214866294072957680886457212, −5.55620948851054911034164030240, −5.47929088624892800989972403653, −5.26837913012982876458972703885, −4.81085766092619711605133367371, −4.69407833192904972852518071414, −4.68132313320446542678656033379, −4.32228864671638402418084076242, −4.00067505904214859449536790195, −3.72602480215257132495847754023, −3.61885296998006644548725224888, −3.60068685911881274103624927916, −2.55415962484326028358638140084, −2.53304099209962704796509945757, −2.42410431095304998578483844774, −1.69605837434562825066105169149, −1.65005308206818863920438839104, −1.47071880303995430767020659839, −1.32688143756791562871071688207, −0.15665999798825613374994193693,
0.15665999798825613374994193693, 1.32688143756791562871071688207, 1.47071880303995430767020659839, 1.65005308206818863920438839104, 1.69605837434562825066105169149, 2.42410431095304998578483844774, 2.53304099209962704796509945757, 2.55415962484326028358638140084, 3.60068685911881274103624927916, 3.61885296998006644548725224888, 3.72602480215257132495847754023, 4.00067505904214859449536790195, 4.32228864671638402418084076242, 4.68132313320446542678656033379, 4.69407833192904972852518071414, 4.81085766092619711605133367371, 5.26837913012982876458972703885, 5.47929088624892800989972403653, 5.55620948851054911034164030240, 6.02214866294072957680886457212, 6.19995369563461895392941089814, 6.20775614029697427791815305532, 6.58754683602131486024868992050, 6.86657709383484072232327963276, 7.14135111731970326643039348502
