| L(s) = 1 | − 4·7-s − 2·19-s + 8·25-s + 12·29-s + 12·41-s + 8·43-s − 2·49-s + 12·53-s − 8·61-s + 24·71-s − 20·73-s + 12·89-s + 24·107-s − 12·113-s + 20·121-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.458·19-s + 8/5·25-s + 2.22·29-s + 1.87·41-s + 1.21·43-s − 2/7·49-s + 1.64·53-s − 1.02·61-s + 2.84·71-s − 2.34·73-s + 1.27·89-s + 2.32·107-s − 1.12·113-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.237569389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237569389\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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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
−8.900840881034435884539232854566, −8.862011073659789189859265964456, −8.247565901398626653927898346463, −7.968607039488244870944908177974, −7.27280471617382105116998514886, −7.23262651496418731159759532417, −6.56491759360658942569256961787, −6.39907636062997152988100109556, −6.11957575570286189759501682185, −5.65670085924131563834963298536, −5.00652095774585820789895156975, −4.78561051033238522822154556347, −4.12270903404328261878565353584, −3.97679949688989378860562759644, −3.06769429339843665983831725370, −3.03465511737228209844666263977, −2.56595461323032056866578078774, −1.92858717990471406488940216022, −0.932692176303675529087860756415, −0.62343445286279761208255850052,
0.62343445286279761208255850052, 0.932692176303675529087860756415, 1.92858717990471406488940216022, 2.56595461323032056866578078774, 3.03465511737228209844666263977, 3.06769429339843665983831725370, 3.97679949688989378860562759644, 4.12270903404328261878565353584, 4.78561051033238522822154556347, 5.00652095774585820789895156975, 5.65670085924131563834963298536, 6.11957575570286189759501682185, 6.39907636062997152988100109556, 6.56491759360658942569256961787, 7.23262651496418731159759532417, 7.27280471617382105116998514886, 7.968607039488244870944908177974, 8.247565901398626653927898346463, 8.862011073659789189859265964456, 8.900840881034435884539232854566
