| L(s) = 1 | − 2-s − 4-s + 3·8-s − 6·9-s − 16-s + 2·17-s + 6·18-s − 8·19-s − 6·25-s − 5·32-s − 2·34-s + 6·36-s + 8·38-s − 12·41-s + 8·43-s + 2·49-s + 6·50-s − 24·59-s + 7·64-s + 8·67-s − 2·68-s − 18·72-s − 12·73-s + 8·76-s + 27·81-s + 12·82-s − 8·83-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 2·9-s − 1/4·16-s + 0.485·17-s + 1.41·18-s − 1.83·19-s − 6/5·25-s − 0.883·32-s − 0.342·34-s + 36-s + 1.29·38-s − 1.87·41-s + 1.21·43-s + 2/7·49-s + 0.848·50-s − 3.12·59-s + 7/8·64-s + 0.977·67-s − 0.242·68-s − 2.12·72-s − 1.40·73-s + 0.917·76-s + 3·81-s + 1.32·82-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.71734099889110000433050476633, −10.15301798570319848347903545855, −9.446849696394434590254762832260, −8.868215889831003694053677970209, −8.695686711870280818326611584019, −7.86052886694471390828228448204, −7.81910395523808377988054564239, −6.64458936246770473314630815851, −6.04868829414574942081046284670, −5.50246131433375739774113763395, −4.74199315541377008016560012773, −3.94486422776053801827158365621, −3.06428895368232173582353504603, −1.98635588682742520402514989313, 0,
1.98635588682742520402514989313, 3.06428895368232173582353504603, 3.94486422776053801827158365621, 4.74199315541377008016560012773, 5.50246131433375739774113763395, 6.04868829414574942081046284670, 6.64458936246770473314630815851, 7.81910395523808377988054564239, 7.86052886694471390828228448204, 8.695686711870280818326611584019, 8.868215889831003694053677970209, 9.446849696394434590254762832260, 10.15301798570319848347903545855, 10.71734099889110000433050476633
