| L(s) = 1 | + 148·4-s + 1.11e3·7-s + 1.73e4·13-s + 1.63e4·16-s − 1.29e5·19-s + 3.14e4·25-s + 1.65e5·28-s − 4.59e5·31-s − 2.16e6·37-s + 9.30e5·43-s + 1.95e6·49-s + 2.56e6·52-s + 2.75e5·61-s + 4.03e6·64-s + 6.28e5·67-s + 1.06e7·73-s − 1.92e7·76-s − 2.20e6·79-s + 1.93e7·91-s + 5.95e6·97-s + 4.64e6·100-s + 1.17e7·103-s + 3.57e7·109-s + 1.83e7·112-s + 1.67e7·121-s − 6.80e7·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·4-s + 1.23·7-s + 2.18·13-s + 16-s − 4.34·19-s + 0.401·25-s + 1.42·28-s − 2.77·31-s − 7.02·37-s + 1.78·43-s + 2.37·49-s + 2.53·52-s + 0.155·61-s + 1.92·64-s + 0.255·67-s + 3.21·73-s − 5.02·76-s − 0.502·79-s + 2.69·91-s + 0.662·97-s + 0.464·100-s + 1.05·103-s + 2.64·109-s + 1.23·112-s + 0.857·121-s − 3.20·124-s − 5.35·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43046721 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.254348399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.254348399\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 37 p^{2} T^{2} + 345 p^{4} T^{4} - 37 p^{16} T^{6} + p^{28} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 31402 T^{2} - 5117430021 T^{4} - 31402 p^{14} T^{6} + p^{28} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 559 T - 511062 T^{2} - 559 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 16713814 T^{2} - 100398255156645 T^{4} - 16713814 p^{14} T^{6} + p^{28} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 667 p T + 73596 p^{2} T^{2} - 667 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 189231154 T^{2} + p^{14} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 32461 T + p^{7} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 18081694 T^{2} - 11592509376880840173 T^{4} - 18081694 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 9599936170 T^{2} - \)\(20\!\cdots\!81\)\( T^{4} - 9599936170 p^{14} T^{6} + p^{28} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 229892 T + 25337717553 T^{2} + 229892 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 541177 T + p^{7} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 264543016114 T^{2} + \)\(32\!\cdots\!35\)\( T^{4} - 264543016114 p^{14} T^{6} + p^{28} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 465112 T - 55489438563 T^{2} - 465112 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 323393423854 T^{2} - \)\(15\!\cdots\!53\)\( T^{4} - 323393423854 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 1296296189242 T^{2} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 4360664458486 T^{2} + \)\(12\!\cdots\!35\)\( T^{4} - 4360664458486 p^{14} T^{6} + p^{28} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 137773 T - 3123761436492 T^{2} - 137773 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 314041 T - 5962089855642 T^{2} - 314041 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 10295330331310 T^{2} + p^{14} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 36569 p T + p^{7} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 1101815 T - 17989912691934 T^{2} + 1101815 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 17317361610166 T^{2} - \)\(43\!\cdots\!73\)\( T^{4} - 17317361610166 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 77662517197858 T^{2} + p^{14} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 2979379 T - 71921585252472 T^{2} - 2979379 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−8.939625542184447772716072308955, −8.897092064082042586569283525489, −8.611976445317705237727675699992, −8.314051730312902350736400734959, −8.094666481790539354377175136529, −7.44955197982964828634801437735, −7.27118321402318010562454204617, −6.92752767827356678399790578435, −6.55684385941575001368145094602, −6.50375985396780860826174541370, −6.01866816778811392137295491902, −5.61355252730637497849978464752, −5.22729657407925212140296226193, −5.20289791449093354253297837794, −4.42826214899664373649937632833, −4.00314312759335155632149631482, −3.75021902064097791309476940041, −3.57815690884264859187221659205, −3.03730464417207867033644700522, −2.02546119988824127141718557448, −1.98244590611665525278405774634, −1.92733741506963746283395496947, −1.59587811819071739509135084825, −0.57022887522433816093147511514, −0.49488333448557701448177495044,
0.49488333448557701448177495044, 0.57022887522433816093147511514, 1.59587811819071739509135084825, 1.92733741506963746283395496947, 1.98244590611665525278405774634, 2.02546119988824127141718557448, 3.03730464417207867033644700522, 3.57815690884264859187221659205, 3.75021902064097791309476940041, 4.00314312759335155632149631482, 4.42826214899664373649937632833, 5.20289791449093354253297837794, 5.22729657407925212140296226193, 5.61355252730637497849978464752, 6.01866816778811392137295491902, 6.50375985396780860826174541370, 6.55684385941575001368145094602, 6.92752767827356678399790578435, 7.27118321402318010562454204617, 7.44955197982964828634801437735, 8.094666481790539354377175136529, 8.314051730312902350736400734959, 8.611976445317705237727675699992, 8.897092064082042586569283525489, 8.939625542184447772716072308955
