| L(s) = 1 | + 2·3-s − 4·5-s + 3·9-s + 4·13-s − 8·15-s + 11·25-s + 4·27-s + 8·31-s − 12·37-s + 8·39-s + 20·43-s − 12·45-s − 49-s − 28·53-s − 16·65-s − 20·67-s + 24·71-s + 22·75-s + 28·79-s + 5·81-s + 24·83-s + 16·89-s + 16·93-s + 8·107-s − 24·111-s + 12·117-s + 6·121-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 9-s + 1.10·13-s − 2.06·15-s + 11/5·25-s + 0.769·27-s + 1.43·31-s − 1.97·37-s + 1.28·39-s + 3.04·43-s − 1.78·45-s − 1/7·49-s − 3.84·53-s − 1.98·65-s − 2.44·67-s + 2.84·71-s + 2.54·75-s + 3.15·79-s + 5/9·81-s + 2.63·83-s + 1.69·89-s + 1.65·93-s + 0.773·107-s − 2.27·111-s + 1.10·117-s + 6/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.020632629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.020632629\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.567819942799249714602010885128, −8.426336635636305127068186430565, −8.098669702489921424113953329183, −7.68130236284686718342587077275, −7.55835377641132013487815061062, −7.19264089521570009848293430474, −6.48689932135419456135463669843, −6.34892621300295132023060141861, −6.05003892972168458098315915763, −5.06674516919571137340226387986, −4.81489010576000963676604277736, −4.61824952685902356533292830462, −3.78978205021579893172733102290, −3.76595964921642079534572064373, −3.41455003614794314970890911207, −2.95233662161984075967218719510, −2.41801035713232971077379923375, −1.81906811780195090324792399110, −1.10170041188640126478914951421, −0.56381971390923017164732230783,
0.56381971390923017164732230783, 1.10170041188640126478914951421, 1.81906811780195090324792399110, 2.41801035713232971077379923375, 2.95233662161984075967218719510, 3.41455003614794314970890911207, 3.76595964921642079534572064373, 3.78978205021579893172733102290, 4.61824952685902356533292830462, 4.81489010576000963676604277736, 5.06674516919571137340226387986, 6.05003892972168458098315915763, 6.34892621300295132023060141861, 6.48689932135419456135463669843, 7.19264089521570009848293430474, 7.55835377641132013487815061062, 7.68130236284686718342587077275, 8.098669702489921424113953329183, 8.426336635636305127068186430565, 8.567819942799249714602010885128
