| L(s) = 1 | + 6·9-s − 4·17-s − 6·25-s + 20·41-s − 14·49-s + 12·73-s + 27·81-s − 20·89-s − 36·97-s − 28·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·9-s − 0.970·17-s − 6/5·25-s + 3.12·41-s − 2·49-s + 1.40·73-s + 3·81-s − 2.11·89-s − 3.65·97-s − 2.63·113-s + 2·121-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-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
| \(L(1)\) |
\(\approx\) |
\(1.215372628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215372628\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−13.53783374545753250377939509110, −13.02824755955478473580174590461, −12.51293411533616317032473602234, −12.41930941597087059874329181038, −11.36090455256956921638940592955, −11.11911838669299626302245413709, −10.51740556902965316698962093181, −9.803680106827546855846298258804, −9.576316991150715064482865075460, −9.061941896594789261995051039125, −8.036557796973512610753317860725, −7.79814902851498306451719208502, −7.01704861305086744721648990389, −6.62189904477336838247127256571, −5.89147895045870161977741553862, −5.01237511108207605464128768470, −4.20531702035781344901108809042, −3.97199102139077686237606533667, −2.57482920414217144195647823211, −1.52019857626457482641329780176,
1.52019857626457482641329780176, 2.57482920414217144195647823211, 3.97199102139077686237606533667, 4.20531702035781344901108809042, 5.01237511108207605464128768470, 5.89147895045870161977741553862, 6.62189904477336838247127256571, 7.01704861305086744721648990389, 7.79814902851498306451719208502, 8.036557796973512610753317860725, 9.061941896594789261995051039125, 9.576316991150715064482865075460, 9.803680106827546855846298258804, 10.51740556902965316698962093181, 11.11911838669299626302245413709, 11.36090455256956921638940592955, 12.41930941597087059874329181038, 12.51293411533616317032473602234, 13.02824755955478473580174590461, 13.53783374545753250377939509110
