| L(s) = 1 | + 872·3-s + 1.84e4·5-s + 1.10e5·7-s + 5.80e5·9-s + 1.64e7·11-s + 1.87e7·13-s + 1.61e7·15-s − 1.53e8·17-s − 1.18e8·19-s + 9.67e7·21-s + 7.18e8·23-s − 1.72e9·25-s + 1.73e9·27-s + 3.09e8·29-s + 5.76e9·31-s + 1.43e10·33-s + 2.04e9·35-s − 1.16e10·37-s + 1.63e10·39-s + 1.31e9·41-s − 2.95e10·43-s + 1.07e10·45-s + 1.23e10·47-s − 2.68e10·49-s − 1.34e11·51-s − 3.80e10·53-s + 3.04e11·55-s + ⋯ |
| L(s) = 1 | + 0.690·3-s + 0.528·5-s + 0.356·7-s + 0.364·9-s + 2.80·11-s + 1.07·13-s + 0.365·15-s − 1.54·17-s − 0.579·19-s + 0.246·21-s + 1.01·23-s − 1.41·25-s + 0.864·27-s + 0.0965·29-s + 1.16·31-s + 1.93·33-s + 0.188·35-s − 0.744·37-s + 0.743·39-s + 0.0431·41-s − 0.713·43-s + 0.192·45-s + 0.166·47-s − 0.277·49-s − 1.06·51-s − 0.235·53-s + 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(4.120005402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.120005402\) |
| \(L(\frac{15}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 872 T + 6658 p^{3} T^{2} - 872 p^{13} T^{3} + p^{26} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 18476 T + 82643774 p^{2} T^{2} - 18476 p^{13} T^{3} + p^{26} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 110928 T + 5599420418 p T^{2} - 110928 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 1497640 p T + 12362903299666 p T^{2} - 1497640 p^{14} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 18744572 T + 344120051417598 T^{2} - 18744572 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9046684 p T + 21058111940247014 T^{2} + 9046684 p^{14} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 118747640 T + 59473815443581782 T^{2} + 118747640 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 718268912 T + 1097306915486653358 T^{2} - 718268912 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 309341340 T + 2504177667132675934 T^{2} - 309341340 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5767504192 T + 50116253247327696702 T^{2} - 5767504192 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11621553300 T + \)\(18\!\cdots\!38\)\( T^{2} + 11621553300 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1311168276 T + \)\(16\!\cdots\!30\)\( T^{2} - 1311168276 p^{13} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 29595620104 T + \)\(33\!\cdots\!74\)\( T^{2} + 29595620104 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12313617888 T + \)\(10\!\cdots\!90\)\( T^{2} - 12313617888 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 38006007028 T + \)\(44\!\cdots\!42\)\( T^{2} + 38006007028 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 253345911704 T + \)\(16\!\cdots\!06\)\( T^{2} - 253345911704 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 647244384292 T + \)\(42\!\cdots\!62\)\( T^{2} + 647244384292 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1619993806312 T + \)\(17\!\cdots\!86\)\( T^{2} - 1619993806312 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1040270142512 T + \)\(22\!\cdots\!82\)\( T^{2} + 1040270142512 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4005283908692 T + \)\(73\!\cdots\!18\)\( T^{2} - 4005283908692 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2521777572064 T + \)\(10\!\cdots\!58\)\( T^{2} + 2521777572064 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 290486230904 T + \)\(14\!\cdots\!54\)\( T^{2} + 290486230904 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8723755657740 T + \)\(61\!\cdots\!42\)\( T^{2} + 8723755657740 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9601712299972 T + \)\(81\!\cdots\!50\)\( T^{2} - 9601712299972 p^{13} T^{3} + p^{26} T^{4} \) |
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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
−19.06146995850490333695964030123, −18.08078359003829456613516026548, −17.27443124754974610040853437638, −16.98880467838786691709757214886, −15.68689867108623463009891937997, −15.10450496132002133661116418687, −13.99239012874510052413366315131, −13.96475608267364780696209232708, −12.84418621935524834943199937424, −11.70940289086481405015905070995, −11.10167734395663192568158008963, −9.784037035096331188508543616076, −8.928015093795377593754458071171, −8.500353796967692721585699993623, −6.77693736343220153026919877280, −6.29808790648200939985756644028, −4.46332533680767050712165009301, −3.61491182189509783853227786530, −1.98847987996368312376160847260, −1.14020365598740027156905769614,
1.14020365598740027156905769614, 1.98847987996368312376160847260, 3.61491182189509783853227786530, 4.46332533680767050712165009301, 6.29808790648200939985756644028, 6.77693736343220153026919877280, 8.500353796967692721585699993623, 8.928015093795377593754458071171, 9.784037035096331188508543616076, 11.10167734395663192568158008963, 11.70940289086481405015905070995, 12.84418621935524834943199937424, 13.96475608267364780696209232708, 13.99239012874510052413366315131, 15.10450496132002133661116418687, 15.68689867108623463009891937997, 16.98880467838786691709757214886, 17.27443124754974610040853437638, 18.08078359003829456613516026548, 19.06146995850490333695964030123
