| L(s) = 1 | − 20·13-s − 2·25-s + 8·37-s + 28·49-s + 8·61-s − 32·73-s − 8·97-s + 40·109-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 198·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 5.54·13-s − 2/5·25-s + 1.31·37-s + 4·49-s + 1.02·61-s − 3.74·73-s − 0.812·97-s + 3.83·109-s − 3.45·121-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 + 15.2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7926854402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7926854402\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.42639464090257224362096110031, −6.15956547338578833517903868716, −5.99004628223642233969274527716, −5.77878425001991175686669282502, −5.62154293604148583925430278153, −5.34438104874535338444917183643, −5.05162141564394852685332829476, −5.04776956249746308902006493515, −4.67876519444255430857859756633, −4.60842711787946228282542843402, −4.48745386809439008593739327482, −4.06615573386152862034482968478, −3.97072032719299627320231334092, −3.69586874581023242496531991373, −3.31348321307895913107532656244, −2.87262717924069307700139240477, −2.78799980749226680917202734656, −2.46744624116600037250749210666, −2.42833168088655551924914365426, −2.27770644008203055420211636656, −1.99314528461951552827039882426, −1.42095124370104356387348182277, −1.11998333996967552298160488249, −0.49965363800857599949030033960, −0.23238267927961133255810148763,
0.23238267927961133255810148763, 0.49965363800857599949030033960, 1.11998333996967552298160488249, 1.42095124370104356387348182277, 1.99314528461951552827039882426, 2.27770644008203055420211636656, 2.42833168088655551924914365426, 2.46744624116600037250749210666, 2.78799980749226680917202734656, 2.87262717924069307700139240477, 3.31348321307895913107532656244, 3.69586874581023242496531991373, 3.97072032719299627320231334092, 4.06615573386152862034482968478, 4.48745386809439008593739327482, 4.60842711787946228282542843402, 4.67876519444255430857859756633, 5.04776956249746308902006493515, 5.05162141564394852685332829476, 5.34438104874535338444917183643, 5.62154293604148583925430278153, 5.77878425001991175686669282502, 5.99004628223642233969274527716, 6.15956547338578833517903868716, 6.42639464090257224362096110031
