L(s) = 1 | − 8·7-s − 2·9-s − 4·17-s + 8·23-s − 6·25-s − 12·41-s − 16·47-s + 34·49-s + 16·63-s + 24·71-s + 28·73-s − 16·79-s − 5·81-s − 4·89-s − 4·97-s − 8·103-s + 4·113-s + 32·119-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + 157-s + ⋯ |
L(s) = 1 | − 3.02·7-s − 2/3·9-s − 0.970·17-s + 1.66·23-s − 6/5·25-s − 1.87·41-s − 2.33·47-s + 34/7·49-s + 2.01·63-s + 2.84·71-s + 3.27·73-s − 1.80·79-s − 5/9·81-s − 0.423·89-s − 0.406·97-s − 0.788·103-s + 0.376·113-s + 2.93·119-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.646·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.90944990278511493765148592647, −9.886348706897054809183790479328, −9.742591354048855107896325616652, −9.385355573475711088007535446099, −8.635939205168057076823393046651, −8.173183686627627779120693205031, −6.97464122003923989664965494039, −6.78571919296342801026858814862, −6.33050260025483473718379487161, −5.64130249043000575312616744005, −4.84373431553687523132526670248, −3.56276783709589056726776122484, −3.36730269661969347724618284285, −2.46608091328452001690059553408, 0,
2.46608091328452001690059553408, 3.36730269661969347724618284285, 3.56276783709589056726776122484, 4.84373431553687523132526670248, 5.64130249043000575312616744005, 6.33050260025483473718379487161, 6.78571919296342801026858814862, 6.97464122003923989664965494039, 8.173183686627627779120693205031, 8.635939205168057076823393046651, 9.385355573475711088007535446099, 9.742591354048855107896325616652, 9.886348706897054809183790479328, 10.90944990278511493765148592647