| L(s) = 1 | + 29·9-s + 78·11-s − 302·19-s − 384·29-s − 36·31-s + 458·41-s + 682·49-s + 672·59-s + 1.71e3·61-s − 1.56e3·71-s + 460·79-s + 112·81-s + 2.73e3·89-s + 2.26e3·99-s − 1.58e3·101-s − 892·109-s + 1.90e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.66e3·169-s + ⋯ |
| L(s) = 1 | + 1.07·9-s + 2.13·11-s − 3.64·19-s − 2.45·29-s − 0.208·31-s + 1.74·41-s + 1.98·49-s + 1.48·59-s + 3.60·61-s − 2.60·71-s + 0.655·79-s + 0.153·81-s + 3.26·89-s + 2.29·99-s − 1.56·101-s − 0.783·109-s + 1.42·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.21·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.491670786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.491670786\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 29 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 39 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2662 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6105 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 151 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20970 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 192 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 82262 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 229 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 132118 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 162702 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 36330 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 336 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 858 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 557845 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 780 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 615625 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 230 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 528275 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1369 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1679422 T^{2} + p^{6} T^{4} \) |
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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
−12.21664126686343460280380840668, −11.86327057327594977330295088102, −11.17815172298845182978276133590, −10.82398019709257388456975935528, −10.38966734222612827932544969790, −9.688306202194238358372953444272, −9.288588163093990104799630600539, −8.721423973935825382515422517497, −8.508724284583640195325111002257, −7.53328298486409843140431375351, −7.05388151539442564828460593270, −6.61292741544722798605061141732, −6.11623984890470599876332336174, −5.51213140444168034112283053022, −4.35030154625172555224361646761, −4.00341493510927220721909268805, −3.87665972532535059145654677602, −2.22846710100585807929670257611, −1.84081017402116021465759854740, −0.70044957857958764887523174408,
0.70044957857958764887523174408, 1.84081017402116021465759854740, 2.22846710100585807929670257611, 3.87665972532535059145654677602, 4.00341493510927220721909268805, 4.35030154625172555224361646761, 5.51213140444168034112283053022, 6.11623984890470599876332336174, 6.61292741544722798605061141732, 7.05388151539442564828460593270, 7.53328298486409843140431375351, 8.508724284583640195325111002257, 8.721423973935825382515422517497, 9.288588163093990104799630600539, 9.688306202194238358372953444272, 10.38966734222612827932544969790, 10.82398019709257388456975935528, 11.17815172298845182978276133590, 11.86327057327594977330295088102, 12.21664126686343460280380840668
