| L(s) = 1 | − 2·2-s − 3-s + 2·6-s − 2·7-s + 4·8-s + 9-s + 4·14-s − 4·16-s − 2·18-s + 19-s + 2·21-s − 4·24-s + 9·25-s − 27-s − 10·29-s − 2·38-s − 4·41-s − 4·42-s + 2·43-s + 4·48-s − 7·49-s − 18·50-s − 6·53-s + 2·54-s − 8·56-s − 57-s + 20·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 0.816·6-s − 0.755·7-s + 1.41·8-s + 1/3·9-s + 1.06·14-s − 16-s − 0.471·18-s + 0.229·19-s + 0.436·21-s − 0.816·24-s + 9/5·25-s − 0.192·27-s − 1.85·29-s − 0.324·38-s − 0.624·41-s − 0.617·42-s + 0.304·43-s + 0.577·48-s − 49-s − 2.54·50-s − 0.824·53-s + 0.272·54-s − 1.06·56-s − 0.132·57-s + 2.62·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 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
−9.154400121981158804401857607700, −8.356796462076166613653114239196, −8.229137186802152492039747690389, −7.57275523727068494228009509627, −7.02168923314163501160140944529, −6.67888033745802292347432374954, −6.06431608345039650918020965328, −5.38584719084054094519193687543, −4.95110308127903308488554601984, −4.36684095993557906721725458582, −3.62211297309065956970107283013, −3.03768766443635528027342196559, −1.90709651860297644790752209471, −0.973292221297076427696333200312, 0,
0.973292221297076427696333200312, 1.90709651860297644790752209471, 3.03768766443635528027342196559, 3.62211297309065956970107283013, 4.36684095993557906721725458582, 4.95110308127903308488554601984, 5.38584719084054094519193687543, 6.06431608345039650918020965328, 6.67888033745802292347432374954, 7.02168923314163501160140944529, 7.57275523727068494228009509627, 8.229137186802152492039747690389, 8.356796462076166613653114239196, 9.154400121981158804401857607700
