| L(s) = 1 | + 2·9-s − 2·11-s + 8·19-s + 12·29-s + 16·31-s + 12·41-s − 2·49-s + 24·59-s + 4·61-s − 24·71-s − 16·79-s − 5·81-s − 12·89-s − 4·99-s − 36·101-s + 20·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 0.603·11-s + 1.83·19-s + 2.22·29-s + 2.87·31-s + 1.87·41-s − 2/7·49-s + 3.12·59-s + 0.512·61-s − 2.84·71-s − 1.80·79-s − 5/9·81-s − 1.27·89-s − 0.402·99-s − 3.58·101-s + 1.91·109-s + 3/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.715957380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.715957380\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.904619601947231462797622914774, −9.715580398034927405479650009820, −9.554126878613921355380491692879, −8.514919887081907299396917559478, −8.450224077549487878662739886029, −8.204583419709775097730619486966, −7.41231500998221510336887564749, −7.22756490142002398012909377944, −6.82407253323772120423210213616, −6.29833131786111481222644385444, −5.66435842287262031346413980528, −5.53452081378025250626141060079, −4.68887016627501765658936316281, −4.48517842398633277794619898587, −4.08321711411749834182680975996, −3.03417605940130255093621577693, −2.94430395547598527273179454973, −2.35617466104163218299710698300, −1.21163412986945572149062512868, −0.891608224610582192073426463959,
0.891608224610582192073426463959, 1.21163412986945572149062512868, 2.35617466104163218299710698300, 2.94430395547598527273179454973, 3.03417605940130255093621577693, 4.08321711411749834182680975996, 4.48517842398633277794619898587, 4.68887016627501765658936316281, 5.53452081378025250626141060079, 5.66435842287262031346413980528, 6.29833131786111481222644385444, 6.82407253323772120423210213616, 7.22756490142002398012909377944, 7.41231500998221510336887564749, 8.204583419709775097730619486966, 8.450224077549487878662739886029, 8.514919887081907299396917559478, 9.554126878613921355380491692879, 9.715580398034927405479650009820, 9.904619601947231462797622914774
