| L(s) = 1 | + 187·4-s − 1.45e3·9-s + 6.89e3·11-s + 821·16-s + 9.91e4·19-s − 3.63e5·29-s + 6.08e5·31-s − 2.72e5·36-s + 1.26e6·41-s + 1.28e6·44-s + 1.50e6·49-s − 5.74e5·59-s − 5.02e6·61-s − 3.16e6·64-s − 7.49e6·71-s + 1.85e7·76-s + 9.25e6·79-s + 1.59e6·81-s − 3.30e7·89-s − 1.00e7·99-s + 4.26e7·101-s − 5.32e7·109-s − 6.79e7·116-s + 1.64e7·121-s + 1.13e8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.46·4-s − 2/3·9-s + 1.56·11-s + 0.0501·16-s + 3.31·19-s − 2.76·29-s + 3.66·31-s − 0.973·36-s + 2.86·41-s + 2.28·44-s + 1.82·49-s − 0.364·59-s − 2.83·61-s − 1.51·64-s − 2.48·71-s + 4.84·76-s + 2.11·79-s + 1/3·81-s − 4.96·89-s − 1.04·99-s + 4.12·101-s − 3.93·109-s − 4.04·116-s + 0.845·121-s + 5.35·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(8.507586178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.507586178\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 187 T^{2} + 8537 p^{2} T^{4} - 187 p^{14} T^{6} + p^{28} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 1501596 T^{2} + 1118934969958 T^{4} - 1501596 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 3448 T + 9598294 T^{2} - 3448 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 6569708 T^{2} - 355743830020842 T^{4} - 6569708 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 1037293660 T^{2} + 596867761160490758 T^{4} - 1037293660 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 49584 T + 1767259558 T^{2} - 49584 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1250733404 T^{2} - 10802449632029146842 T^{4} - 1250733404 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6268 p T + 32439203278 T^{2} + 6268 p^{8} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 304232 T + 77458297022 T^{2} - 304232 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 191047267916 T^{2} + \)\(19\!\cdots\!78\)\( T^{4} - 191047267916 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 631172 T + 420346017142 T^{2} - 631172 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1009441609580 T^{2} + \)\(40\!\cdots\!98\)\( T^{4} - 1009441609580 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1906720707580 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - 1906720707580 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4484889731884 T^{2} + \)\(77\!\cdots\!78\)\( T^{4} - 4484889731884 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 287224 T + 1627391637238 T^{2} + 287224 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 2514180 T + 7865442419758 T^{2} + 2514180 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6298759315980 T^{2} + \)\(18\!\cdots\!58\)\( T^{4} - 6298759315980 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 3748816 T + 20824804809646 T^{2} + 3748816 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 2188471674916 T^{2} + \)\(19\!\cdots\!78\)\( T^{4} + 2188471674916 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4627720 T + 42789559383518 T^{2} - 4627720 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 87567568581068 T^{2} + \)\(33\!\cdots\!98\)\( T^{4} - 87567568581068 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16516356 T + 156597055746838 T^{2} + 16516356 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 264273066872060 T^{2} + \)\(30\!\cdots\!38\)\( T^{4} - 264273066872060 p^{14} T^{6} + p^{28} T^{8} \) |
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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
−9.224412258582351554814460570829, −9.189504437588433468152821980789, −8.743241885003031377192143312017, −8.338637227380177897345887814873, −7.889982157297160934728662767695, −7.55348418377975892237104618407, −7.29476785008521870096859244249, −7.23954837339629772953412609492, −6.88011051612882776426364561846, −6.18822244596310844680223495004, −6.09302013957562957744706065307, −5.96133831624800437813260982859, −5.57759489937859692685839935414, −4.97839928213605064886518149592, −4.67114196629996571271312502599, −4.07571817419419733292694966849, −3.95383296376084929998164302586, −3.22847817927448865158129898195, −2.93873427928705183862364771014, −2.67572102875463801800715405602, −2.30775824685608283678593078560, −1.43664926949453331242169652643, −1.43511850036141213543608155089, −0.912186486730257612815021736913, −0.42101794811626662211555585857,
0.42101794811626662211555585857, 0.912186486730257612815021736913, 1.43511850036141213543608155089, 1.43664926949453331242169652643, 2.30775824685608283678593078560, 2.67572102875463801800715405602, 2.93873427928705183862364771014, 3.22847817927448865158129898195, 3.95383296376084929998164302586, 4.07571817419419733292694966849, 4.67114196629996571271312502599, 4.97839928213605064886518149592, 5.57759489937859692685839935414, 5.96133831624800437813260982859, 6.09302013957562957744706065307, 6.18822244596310844680223495004, 6.88011051612882776426364561846, 7.23954837339629772953412609492, 7.29476785008521870096859244249, 7.55348418377975892237104618407, 7.889982157297160934728662767695, 8.338637227380177897345887814873, 8.743241885003031377192143312017, 9.189504437588433468152821980789, 9.224412258582351554814460570829
