| L(s) = 1 | − 8·11-s + 4·25-s − 24·43-s − 2·49-s − 40·67-s − 40·107-s − 80·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 4/5·25-s − 3.65·43-s − 2/7·49-s − 4.88·67-s − 3.86·107-s − 7.52·113-s − 0.363·121-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 − 2.15·169-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 + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1744680386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1744680386\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.54176679545598488878060503293, −6.48574624436236406769237590991, −6.06241084241249752537342049884, −5.77814084424305553768198895184, −5.46840278866238273656538974749, −5.46518281367981248751729589091, −5.35092237664243075146146991420, −5.02905759341184778518200530174, −4.79808115639578430152815424944, −4.70962970947631335063800880786, −4.42599728015629921609336796027, −4.05997634665677287921481616420, −3.93856368612285977608286734031, −3.71887702956303727931998851161, −3.18560357978245962740474338725, −3.02250514615957291880803423832, −2.94924452638217018171085095942, −2.72512332589468959144968450107, −2.53356547690894924507116472626, −2.12699708870489255870517230089, −1.54069860859275176556488498341, −1.53232942630085786462168250816, −1.43149161532078904481734151453, −0.53633225587813873199241415417, −0.098648222702785077779240442056,
0.098648222702785077779240442056, 0.53633225587813873199241415417, 1.43149161532078904481734151453, 1.53232942630085786462168250816, 1.54069860859275176556488498341, 2.12699708870489255870517230089, 2.53356547690894924507116472626, 2.72512332589468959144968450107, 2.94924452638217018171085095942, 3.02250514615957291880803423832, 3.18560357978245962740474338725, 3.71887702956303727931998851161, 3.93856368612285977608286734031, 4.05997634665677287921481616420, 4.42599728015629921609336796027, 4.70962970947631335063800880786, 4.79808115639578430152815424944, 5.02905759341184778518200530174, 5.35092237664243075146146991420, 5.46518281367981248751729589091, 5.46840278866238273656538974749, 5.77814084424305553768198895184, 6.06241084241249752537342049884, 6.48574624436236406769237590991, 6.54176679545598488878060503293
