| L(s) = 1 | − 2·5-s − 2·9-s − 12·13-s + 4·17-s + 3·25-s − 4·29-s + 4·37-s − 20·41-s + 4·45-s − 10·49-s + 4·53-s + 4·61-s + 24·65-s + 20·73-s − 5·81-s − 8·85-s − 12·89-s + 20·97-s + 28·101-s − 28·109-s − 12·113-s + 24·117-s − 6·121-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2/3·9-s − 3.32·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s + 0.657·37-s − 3.12·41-s + 0.596·45-s − 1.42·49-s + 0.549·53-s + 0.512·61-s + 2.97·65-s + 2.34·73-s − 5/9·81-s − 0.867·85-s − 1.27·89-s + 2.03·97-s + 2.78·101-s − 2.68·109-s − 1.12·113-s + 2.21·117-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−10.29364010457156446481778884088, −9.759510932768572667567645779831, −9.572587619444812153159504270402, −8.731345173958242569557384315653, −7.967338729692305876626971946668, −7.85140754717246485615619273252, −7.05269675268361745315034445294, −6.81951269321906735272373937416, −5.70243463207358237884993422463, −4.92945525258561389220988501216, −4.88847555485350104575427844162, −3.69518610983128083383746206280, −3.01064345970677263239408332098, −2.17341817164904427445574627486, 0,
2.17341817164904427445574627486, 3.01064345970677263239408332098, 3.69518610983128083383746206280, 4.88847555485350104575427844162, 4.92945525258561389220988501216, 5.70243463207358237884993422463, 6.81951269321906735272373937416, 7.05269675268361745315034445294, 7.85140754717246485615619273252, 7.967338729692305876626971946668, 8.731345173958242569557384315653, 9.572587619444812153159504270402, 9.759510932768572667567645779831, 10.29364010457156446481778884088
