| L(s) = 1 | − 4·3-s − 4·7-s + 8·9-s + 16·21-s − 12·27-s + 8·31-s − 24·37-s + 24·43-s + 8·49-s − 24·61-s − 32·63-s − 16·67-s + 20·73-s + 23·81-s − 32·93-s − 12·97-s + 4·103-s + 96·111-s + 40·121-s + 127-s − 96·129-s + 131-s + 137-s + 139-s − 32·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.51·7-s + 8/3·9-s + 3.49·21-s − 2.30·27-s + 1.43·31-s − 3.94·37-s + 3.65·43-s + 8/7·49-s − 3.07·61-s − 4.03·63-s − 1.95·67-s + 2.34·73-s + 23/9·81-s − 3.31·93-s − 1.21·97-s + 0.394·103-s + 9.11·111-s + 3.63·121-s + 0.0887·127-s − 8.45·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3907223470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3907223470\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 170 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.05396175211584730588091746628, −6.51857339768752442214269681994, −6.41993687001169517417098489733, −6.23981948073651977075704086946, −6.09672682440284527550956039905, −5.89549984311615117020681657148, −5.87874776283716176884648340049, −5.30193416264812846710918154016, −5.26771250884619389836909941152, −5.04128471302963264032310056062, −4.85610762069170490932897964477, −4.45345377798473796243884228350, −4.37813408933350419936189199090, −4.03916316560390462091312701784, −3.54602120100338245685285433368, −3.51554869002160709638980914341, −3.46016032193849523947999762309, −2.77733370164046151994765170182, −2.60046458156208062329972334724, −2.43395813138501739073090722023, −1.85912937257767425104579989893, −1.32378838970791182785476575744, −1.30091233928686285794980887182, −0.52114991403707118090231917468, −0.29118984894164705980880808373,
0.29118984894164705980880808373, 0.52114991403707118090231917468, 1.30091233928686285794980887182, 1.32378838970791182785476575744, 1.85912937257767425104579989893, 2.43395813138501739073090722023, 2.60046458156208062329972334724, 2.77733370164046151994765170182, 3.46016032193849523947999762309, 3.51554869002160709638980914341, 3.54602120100338245685285433368, 4.03916316560390462091312701784, 4.37813408933350419936189199090, 4.45345377798473796243884228350, 4.85610762069170490932897964477, 5.04128471302963264032310056062, 5.26771250884619389836909941152, 5.30193416264812846710918154016, 5.87874776283716176884648340049, 5.89549984311615117020681657148, 6.09672682440284527550956039905, 6.23981948073651977075704086946, 6.41993687001169517417098489733, 6.51857339768752442214269681994, 7.05396175211584730588091746628
