| L(s) = 1 | + 4·7-s + 56·19-s + 172·25-s − 240·29-s − 320·31-s + 392·37-s − 816·47-s − 288·53-s − 1.82e3·59-s + 1.68e3·83-s − 3.58e3·103-s − 2.62e3·109-s − 4.70e3·113-s + 7.68e3·121-s + 127-s + 131-s + 224·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.73e3·169-s + 173-s + 688·175-s + ⋯ |
| L(s) = 1 | + 0.215·7-s + 0.676·19-s + 1.37·25-s − 1.53·29-s − 1.85·31-s + 1.74·37-s − 2.53·47-s − 0.746·53-s − 4.02·59-s + 2.22·83-s − 3.42·103-s − 2.30·109-s − 3.91·113-s + 5.77·121-s + 0.000698·127-s + 0.000666·131-s + 0.146·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.52·169-s + 0.000439·173-s + 0.297·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.193320575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.193320575\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 4 T + 16 T^{2} + 436 p T^{3} - 3490 p^{2} T^{4} + 436 p^{4} T^{5} + 16 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \) |
| good | 5 | \( 1 - 172 T^{2} + 12088 T^{4} - 1922884 T^{6} + 432877966 T^{8} - 1922884 p^{6} T^{10} + 12088 p^{12} T^{12} - 172 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( 1 - 7684 T^{2} + 29103112 T^{4} - 68711357356 T^{6} + 109991929594990 T^{8} - 68711357356 p^{6} T^{10} + 29103112 p^{12} T^{12} - 7684 p^{18} T^{14} + p^{24} T^{16} \) |
| 13 | \( 1 - 7736 T^{2} + 33258844 T^{4} - 103335049928 T^{6} + 251087208174118 T^{8} - 103335049928 p^{6} T^{10} + 33258844 p^{12} T^{12} - 7736 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( 1 - 25708 T^{2} + 341684344 T^{4} - 2888689279012 T^{6} + 16925607763770286 T^{8} - 2888689279012 p^{6} T^{10} + 341684344 p^{12} T^{12} - 25708 p^{18} T^{14} + p^{24} T^{16} \) |
| 19 | \( ( 1 - 28 T + 17056 T^{2} - 75836 T^{3} + 133946894 T^{4} - 75836 p^{3} T^{5} + 17056 p^{6} T^{6} - 28 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 23 | \( 1 - 29476 T^{2} + 511325128 T^{4} - 7961603563564 T^{6} + 110519492921771470 T^{8} - 7961603563564 p^{6} T^{10} + 511325128 p^{12} T^{12} - 29476 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( ( 1 + 120 T + 65660 T^{2} + 6712488 T^{3} + 2255693046 T^{4} + 6712488 p^{3} T^{5} + 65660 p^{6} T^{6} + 120 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 160 T + 36172 T^{2} - 6186976 T^{3} - 728821786 T^{4} - 6186976 p^{3} T^{5} + 36172 p^{6} T^{6} + 160 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 196 T + 70696 T^{2} - 15303292 T^{3} + 2331128254 T^{4} - 15303292 p^{3} T^{5} + 70696 p^{6} T^{6} - 196 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 41 | \( 1 - 237772 T^{2} + 34664599288 T^{4} - 3564171608723332 T^{6} + \)\(28\!\cdots\!46\)\( T^{8} - 3564171608723332 p^{6} T^{10} + 34664599288 p^{12} T^{12} - 237772 p^{18} T^{14} + p^{24} T^{16} \) |
| 43 | \( 1 - 273872 T^{2} + 34345238716 T^{4} - 2691741399065264 T^{6} + \)\(18\!\cdots\!22\)\( T^{8} - 2691741399065264 p^{6} T^{10} + 34345238716 p^{12} T^{12} - 273872 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( ( 1 + 408 T + 266828 T^{2} + 62051256 T^{3} + 32091577638 T^{4} + 62051256 p^{3} T^{5} + 266828 p^{6} T^{6} + 408 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 + 144 T + 272156 T^{2} + 93836016 T^{3} + 36246347862 T^{4} + 93836016 p^{3} T^{5} + 272156 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 + 912 T + 752684 T^{2} + 485643024 T^{3} + 219867046134 T^{4} + 485643024 p^{3} T^{5} + 752684 p^{6} T^{6} + 912 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 61 | \( 1 - 1695272 T^{2} + 1282321093564 T^{4} - 565249437068217176 T^{6} + \)\(15\!\cdots\!66\)\( T^{8} - 565249437068217176 p^{6} T^{10} + 1282321093564 p^{12} T^{12} - 1695272 p^{18} T^{14} + p^{24} T^{16} \) |
| 67 | \( 1 - 1674176 T^{2} + 1368230434588 T^{4} - 709074585714612800 T^{6} + \)\(25\!\cdots\!78\)\( T^{8} - 709074585714612800 p^{6} T^{10} + 1368230434588 p^{12} T^{12} - 1674176 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( 1 - 1345636 T^{2} + 1085567531464 T^{4} - 599054164045845100 T^{6} + \)\(24\!\cdots\!66\)\( T^{8} - 599054164045845100 p^{6} T^{10} + 1085567531464 p^{12} T^{12} - 1345636 p^{18} T^{14} + p^{24} T^{16} \) |
| 73 | \( 1 - 1739480 T^{2} + 1687675343356 T^{4} - 1071753557470214888 T^{6} + \)\(49\!\cdots\!98\)\( T^{8} - 1071753557470214888 p^{6} T^{10} + 1687675343356 p^{12} T^{12} - 1739480 p^{18} T^{14} + p^{24} T^{16} \) |
| 79 | \( 1 - 3231392 T^{2} + 4863579897916 T^{4} - 4425141721050671456 T^{6} + \)\(26\!\cdots\!14\)\( T^{8} - 4425141721050671456 p^{6} T^{10} + 4863579897916 p^{12} T^{12} - 3231392 p^{18} T^{14} + p^{24} T^{16} \) |
| 83 | \( ( 1 - 840 T + 2002748 T^{2} - 1107399240 T^{3} + 1578867394614 T^{4} - 1107399240 p^{3} T^{5} + 2002748 p^{6} T^{6} - 840 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 89 | \( 1 - 3966028 T^{2} + 7744575214072 T^{4} - 9543109070842050436 T^{6} + \)\(80\!\cdots\!46\)\( T^{8} - 9543109070842050436 p^{6} T^{10} + 7744575214072 p^{12} T^{12} - 3966028 p^{18} T^{14} + p^{24} T^{16} \) |
| 97 | \( 1 - 3256088 T^{2} + 6152309237692 T^{4} - 8283569126499185960 T^{6} + \)\(84\!\cdots\!90\)\( T^{8} - 8283569126499185960 p^{6} T^{10} + 6152309237692 p^{12} T^{12} - 3256088 p^{18} T^{14} + p^{24} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.90789699404240530316055326775, −3.70031497555661596419507821099, −3.59311122062093389255980754618, −3.28114717916806850324684630524, −3.21139830772956491784043862457, −3.18096515180764721712761268345, −3.10235580004682165874305488680, −3.06089594531362594923959344490, −2.74420425906499074818310916376, −2.53951459909138056351243257116, −2.52489125411924863833667369030, −2.50774821514057327942144500266, −1.88812612979730380031905369560, −1.85319099211412801920272299077, −1.80122431081106946949298577774, −1.78914500997470519147314885813, −1.69997506251133073841352119263, −1.48835524243482334653722756953, −1.17885106174725199314245160975, −0.912296863696480535546254226492, −0.883698263805018606097238158097, −0.73961735875796051187607948212, −0.45955385096986237069524752391, −0.21972886846346508358131368985, −0.092778977568969205774594739992,
0.092778977568969205774594739992, 0.21972886846346508358131368985, 0.45955385096986237069524752391, 0.73961735875796051187607948212, 0.883698263805018606097238158097, 0.912296863696480535546254226492, 1.17885106174725199314245160975, 1.48835524243482334653722756953, 1.69997506251133073841352119263, 1.78914500997470519147314885813, 1.80122431081106946949298577774, 1.85319099211412801920272299077, 1.88812612979730380031905369560, 2.50774821514057327942144500266, 2.52489125411924863833667369030, 2.53951459909138056351243257116, 2.74420425906499074818310916376, 3.06089594531362594923959344490, 3.10235580004682165874305488680, 3.18096515180764721712761268345, 3.21139830772956491784043862457, 3.28114717916806850324684630524, 3.59311122062093389255980754618, 3.70031497555661596419507821099, 3.90789699404240530316055326775
Plot not available for L-functions of degree greater than 10.
