L(s) = 1 | − 2-s − 4·3-s + 4-s + 4·6-s + 7-s − 8-s + 6·9-s − 4·12-s − 14-s + 16-s − 6·18-s + 4·19-s − 4·21-s + 4·24-s − 10·25-s + 4·27-s + 28-s − 12·29-s − 8·31-s − 32-s + 6·36-s + 4·37-s − 4·38-s + 4·42-s − 24·47-s − 4·48-s + 49-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s + 1.63·6-s + 0.377·7-s − 0.353·8-s + 2·9-s − 1.15·12-s − 0.267·14-s + 1/4·16-s − 1.41·18-s + 0.917·19-s − 0.872·21-s + 0.816·24-s − 2·25-s + 0.769·27-s + 0.188·28-s − 2.22·29-s − 1.43·31-s − 0.176·32-s + 36-s + 0.657·37-s − 0.648·38-s + 0.617·42-s − 3.50·47-s − 0.577·48-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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
−11.23136141438460806904855801814, −11.03116828040960387372749944303, −10.02053664953109744946236323249, −9.765547119459919407856461234632, −9.044637307272642732451675861332, −8.199638847074308569053560326614, −7.57571100088867902110310233811, −7.01851678171449461328017406897, −6.24500065333065006116855945568, −5.57928681742950427486583645839, −5.53331414913354333592453982899, −4.55372588498345584946220609716, −3.42554077807776045337947490220, −1.73289286183865826039844070287, 0,
1.73289286183865826039844070287, 3.42554077807776045337947490220, 4.55372588498345584946220609716, 5.53331414913354333592453982899, 5.57928681742950427486583645839, 6.24500065333065006116855945568, 7.01851678171449461328017406897, 7.57571100088867902110310233811, 8.199638847074308569053560326614, 9.044637307272642732451675861332, 9.765547119459919407856461234632, 10.02053664953109744946236323249, 11.03116828040960387372749944303, 11.23136141438460806904855801814