| L(s) = 1 | − 3-s + 3·4-s + 3·5-s − 7-s − 2·9-s − 3·12-s − 3·15-s + 5·16-s − 6·19-s + 9·20-s + 21-s − 3·23-s − 25-s + 5·27-s − 3·28-s − 3·35-s − 6·36-s − 8·43-s − 6·45-s − 5·48-s − 6·49-s − 21·53-s + 6·57-s − 9·60-s − 3·61-s + 2·63-s + 3·64-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 3/2·4-s + 1.34·5-s − 0.377·7-s − 2/3·9-s − 0.866·12-s − 0.774·15-s + 5/4·16-s − 1.37·19-s + 2.01·20-s + 0.218·21-s − 0.625·23-s − 1/5·25-s + 0.962·27-s − 0.566·28-s − 0.507·35-s − 36-s − 1.21·43-s − 0.894·45-s − 0.721·48-s − 6/7·49-s − 2.88·53-s + 0.794·57-s − 1.16·60-s − 0.384·61-s + 0.251·63-s + 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 815409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 815409 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 43 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 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
−7.974550980322902347333131164516, −7.60383904157420714797236350844, −6.70404021830224289662454654631, −6.57511730371682693498855323303, −6.19341001071073607162998959062, −6.12325847974528321005050259188, −5.26031347413927630316659571989, −5.19203377064154276153616766669, −4.32543983788932883358558201516, −3.57483789292912678147173477207, −3.03626379877431220801689551725, −2.39412421090999333338756546064, −2.01324341630670282407992249861, −1.47280314813167772765949348816, 0,
1.47280314813167772765949348816, 2.01324341630670282407992249861, 2.39412421090999333338756546064, 3.03626379877431220801689551725, 3.57483789292912678147173477207, 4.32543983788932883358558201516, 5.19203377064154276153616766669, 5.26031347413927630316659571989, 6.12325847974528321005050259188, 6.19341001071073607162998959062, 6.57511730371682693498855323303, 6.70404021830224289662454654631, 7.60383904157420714797236350844, 7.974550980322902347333131164516
