| L(s) = 1 | − 2·2-s − 4-s + 8·8-s − 6·9-s − 4·13-s − 7·16-s + 17-s + 12·18-s − 8·19-s − 6·25-s + 8·26-s − 14·32-s − 2·34-s + 6·36-s + 16·38-s + 8·43-s + 2·49-s + 12·50-s + 4·52-s + 12·53-s − 24·59-s + 35·64-s + 8·67-s − 68-s − 48·72-s + 8·76-s + 27·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 2.82·8-s − 2·9-s − 1.10·13-s − 7/4·16-s + 0.242·17-s + 2.82·18-s − 1.83·19-s − 6/5·25-s + 1.56·26-s − 2.47·32-s − 0.342·34-s + 36-s + 2.59·38-s + 1.21·43-s + 2/7·49-s + 1.69·50-s + 0.554·52-s + 1.64·53-s − 3.12·59-s + 35/8·64-s + 0.977·67-s − 0.121·68-s − 5.65·72-s + 0.917·76-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4913 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4913 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−11.93623114442209313196485845061, −11.02941934245556844339132034347, −10.71734099889110000433050476633, −10.04009807089741531902747917178, −9.478907532433233918366672610314, −8.796016188301955158560715963145, −8.695686711870280818326611584019, −7.81910395523808377988054564239, −7.59089249699874684434940494125, −6.30307949005461764625435345620, −5.50273032332247760535071006049, −4.74199315541377008016560012773, −3.87810234950044466939896282289, −2.34450910132149137086379940903, 0,
2.34450910132149137086379940903, 3.87810234950044466939896282289, 4.74199315541377008016560012773, 5.50273032332247760535071006049, 6.30307949005461764625435345620, 7.59089249699874684434940494125, 7.81910395523808377988054564239, 8.695686711870280818326611584019, 8.796016188301955158560715963145, 9.478907532433233918366672610314, 10.04009807089741531902747917178, 10.71734099889110000433050476633, 11.02941934245556844339132034347, 11.93623114442209313196485845061
