| L(s) = 1 | + 4·5-s − 6·9-s − 12·13-s + 4·17-s + 2·25-s + 20·29-s + 4·37-s + 20·41-s − 24·45-s − 14·49-s − 28·53-s + 20·61-s − 48·65-s − 12·73-s + 27·81-s + 16·85-s + 20·89-s + 36·97-s + 4·101-s − 12·109-s − 28·113-s + 72·117-s − 22·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2·9-s − 3.32·13-s + 0.970·17-s + 2/5·25-s + 3.71·29-s + 0.657·37-s + 3.12·41-s − 3.57·45-s − 2·49-s − 3.84·53-s + 2.56·61-s − 5.95·65-s − 1.40·73-s + 3·81-s + 1.73·85-s + 2.11·89-s + 3.65·97-s + 0.398·101-s − 1.14·109-s − 2.63·113-s + 6.65·117-s − 2·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8593982272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8593982272\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−17.63226515593639093296341049403, −17.33116792549252764742350535793, −17.33116792549252764742350535793, −16.11526573364450409746230288294, −16.11526573364450409746230288294, −14.59250078389593402269438368141, −14.59250078389593402269438368141, −14.09304813859309984769942580998, −14.09304813859309984769942580998, −12.72487277908244699547747964129, −12.72487277908244699547747964129, −11.66649219374340327203675385621, −11.66649219374340327203675385621, −10.23421229877843747435058592930, −10.23421229877843747435058592930, −9.317356556158666156286958675354, −9.317356556158666156286958675354, −7.86574055998535746161422058704, −7.86574055998535746161422058704, −6.27282080646876620831649653268, −6.27282080646876620831649653268, −5.01452297596172644651029655270, −5.01452297596172644651029655270, −2.63895104553179739845027310173, −2.63895104553179739845027310173,
2.63895104553179739845027310173, 2.63895104553179739845027310173, 5.01452297596172644651029655270, 5.01452297596172644651029655270, 6.27282080646876620831649653268, 6.27282080646876620831649653268, 7.86574055998535746161422058704, 7.86574055998535746161422058704, 9.317356556158666156286958675354, 9.317356556158666156286958675354, 10.23421229877843747435058592930, 10.23421229877843747435058592930, 11.66649219374340327203675385621, 11.66649219374340327203675385621, 12.72487277908244699547747964129, 12.72487277908244699547747964129, 14.09304813859309984769942580998, 14.09304813859309984769942580998, 14.59250078389593402269438368141, 14.59250078389593402269438368141, 16.11526573364450409746230288294, 16.11526573364450409746230288294, 17.33116792549252764742350535793, 17.33116792549252764742350535793, 17.63226515593639093296341049403
