| L(s) = 1 | + 2·7-s + 16-s + 2·19-s + 4·31-s − 2·37-s + 2·49-s + 2·67-s − 4·73-s − 2·97-s − 4·109-s + 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2·7-s + 16-s + 2·19-s + 4·31-s − 2·37-s + 2·49-s + 2·67-s − 4·73-s − 2·97-s − 4·109-s + 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.896664605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.896664605\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−7.05664712146481997194851636311, −6.76616438476882759252615104054, −6.50127339193516017243931254201, −6.31474192548659897178514497844, −6.17477414005074460699207809921, −5.67487392834577209733013405263, −5.61173532513933884525156385800, −5.44727529413500105361017666323, −5.19795534094704571975896521050, −5.01389372601615699602133953016, −4.71105790625125623305125133738, −4.64762352851899894365460135792, −4.43119338999987451958108208254, −4.01381583352518620004423396348, −3.83953811371073736117525611212, −3.53576311710847286512695306089, −3.40272903615603880908630079838, −2.80054720559142132140513939114, −2.73496951436913743633077015999, −2.62876736595995232556270387518, −2.27257178374952788949614248738, −1.55421796591522024873930573635, −1.46383854567660906090243976228, −1.24913713137494876501298609548, −1.00590992146060072376580203962,
1.00590992146060072376580203962, 1.24913713137494876501298609548, 1.46383854567660906090243976228, 1.55421796591522024873930573635, 2.27257178374952788949614248738, 2.62876736595995232556270387518, 2.73496951436913743633077015999, 2.80054720559142132140513939114, 3.40272903615603880908630079838, 3.53576311710847286512695306089, 3.83953811371073736117525611212, 4.01381583352518620004423396348, 4.43119338999987451958108208254, 4.64762352851899894365460135792, 4.71105790625125623305125133738, 5.01389372601615699602133953016, 5.19795534094704571975896521050, 5.44727529413500105361017666323, 5.61173532513933884525156385800, 5.67487392834577209733013405263, 6.17477414005074460699207809921, 6.31474192548659897178514497844, 6.50127339193516017243931254201, 6.76616438476882759252615104054, 7.05664712146481997194851636311
