| L(s) = 1 | + 4-s − 2·7-s − 2·9-s − 4·11-s − 4·13-s − 3·16-s + 2·17-s + 16·19-s − 3·23-s + 2·25-s − 2·28-s − 2·36-s − 12·41-s − 4·44-s − 3·49-s − 4·52-s − 8·53-s − 6·61-s + 4·63-s − 7·64-s − 8·67-s + 2·68-s − 8·71-s − 16·73-s + 16·76-s + 8·77-s − 5·81-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 2/3·9-s − 1.20·11-s − 1.10·13-s − 3/4·16-s + 0.485·17-s + 3.67·19-s − 0.625·23-s + 2/5·25-s − 0.377·28-s − 1/3·36-s − 1.87·41-s − 0.603·44-s − 3/7·49-s − 0.554·52-s − 1.09·53-s − 0.768·61-s + 0.503·63-s − 7/8·64-s − 0.977·67-s + 0.242·68-s − 0.949·71-s − 1.87·73-s + 1.83·76-s + 0.911·77-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136367 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136367 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 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
−9.273423639723982018200504114081, −8.660780673925602274626671249390, −7.893197932279197685968839791771, −7.65938933616914470248313997028, −7.19630163253202072666816583560, −6.78500429066829218271376986681, −5.98909058502717914495474447974, −5.55725211028915853921131858801, −5.07922211228536297460948352739, −4.65535705169746737640394609079, −3.41400568577537827306056361668, −3.02382588120613875918599234718, −2.70812253375022481376593745545, −1.54679401159096563775798069886, 0,
1.54679401159096563775798069886, 2.70812253375022481376593745545, 3.02382588120613875918599234718, 3.41400568577537827306056361668, 4.65535705169746737640394609079, 5.07922211228536297460948352739, 5.55725211028915853921131858801, 5.98909058502717914495474447974, 6.78500429066829218271376986681, 7.19630163253202072666816583560, 7.65938933616914470248313997028, 7.893197932279197685968839791771, 8.660780673925602274626671249390, 9.273423639723982018200504114081
