| L(s) = 1 | + 12·5-s + 2·9-s − 560·13-s + 52·17-s + 18·25-s − 272·29-s − 484·37-s − 464·41-s + 24·45-s + 540·49-s − 1.89e3·53-s + 1.50e3·61-s − 6.72e3·65-s + 564·73-s + 1.19e3·81-s + 624·85-s + 1.12e3·89-s + 1.87e3·97-s + 1.88e3·101-s + 3.08e3·109-s + 4.56e3·113-s − 1.12e3·117-s + 3.29e3·121-s − 3.86e3·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.07·5-s + 2/27·9-s − 11.9·13-s + 0.741·17-s + 0.143·25-s − 1.74·29-s − 2.15·37-s − 1.76·41-s + 0.0795·45-s + 1.57·49-s − 4.90·53-s + 3.14·61-s − 12.8·65-s + 0.904·73-s + 1.63·81-s + 0.796·85-s + 1.33·89-s + 1.95·97-s + 1.85·101-s + 2.71·109-s + 3.79·113-s − 0.884·117-s + 2.47·121-s − 2.76·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4798774022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4798774022\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 540 T^{2} + 586 p^{3} T^{4} - 540 p^{6} T^{6} + p^{12} T^{8} \) |
| good | 3 | \( 1 - 2 T^{2} - 1187 T^{4} + 178 p T^{6} + 98236 p^{2} T^{8} + 178 p^{7} T^{10} - 1187 p^{12} T^{12} - 2 p^{18} T^{14} + p^{24} T^{16} \) |
| 5 | \( ( 1 - 6 T + 9 p T^{2} + 1554 T^{3} - 20044 T^{4} + 1554 p^{3} T^{5} + 9 p^{7} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 11 | \( 1 - 3290 T^{2} + 5117653 T^{4} - 7117339250 T^{6} + 10249555566988 T^{8} - 7117339250 p^{6} T^{10} + 5117653 p^{12} T^{12} - 3290 p^{18} T^{14} + p^{24} T^{16} \) |
| 13 | \( ( 1 + 140 T + 9026 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 17 | \( ( 1 - 26 T - 6907 T^{2} + 58318 T^{3} + 30043132 T^{4} + 58318 p^{3} T^{5} - 6907 p^{6} T^{6} - 26 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 19 | \( 1 - 20250 T^{2} + 219565781 T^{4} - 1952200379250 T^{6} + 14783933453136300 T^{8} - 1952200379250 p^{6} T^{10} + 219565781 p^{12} T^{12} - 20250 p^{18} T^{14} + p^{24} T^{16} \) |
| 23 | \( 1 - 21602 T^{2} + 121181293 T^{4} - 1066994779466 T^{6} + 28351103221428844 T^{8} - 1066994779466 p^{6} T^{10} + 121181293 p^{12} T^{12} - 21602 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( ( 1 + 68 T + 4642 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 31 | \( 1 - 4906 T^{2} - 796591403 T^{4} + 4682026985438 T^{6} - 135540736460397956 T^{8} + 4682026985438 p^{6} T^{10} - 796591403 p^{12} T^{12} - 4906 p^{18} T^{14} + p^{24} T^{16} \) |
| 37 | \( ( 1 + 242 T - 56311 T^{2} + 3283698 T^{3} + 7664096924 T^{4} + 3283698 p^{3} T^{5} - 56311 p^{6} T^{6} + 242 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 + 116 T + 134506 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 + 243756 T^{2} + 26424972982 T^{4} + 243756 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( 1 - 57514 T^{2} - 14064428011 T^{4} + 240761796926814 T^{6} + \)\(14\!\cdots\!24\)\( T^{8} + 240761796926814 p^{6} T^{10} - 14064428011 p^{12} T^{12} - 57514 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( ( 1 + 946 T + 390585 T^{2} + 195421842 T^{3} + 98952987100 T^{4} + 195421842 p^{3} T^{5} + 390585 p^{6} T^{6} + 946 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 59 | \( 1 + 87550 T^{2} - 71115411779 T^{4} - 488586170412650 T^{6} + \)\(41\!\cdots\!60\)\( T^{8} - 488586170412650 p^{6} T^{10} - 71115411779 p^{12} T^{12} + 87550 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( ( 1 - 750 T - 31015 T^{2} - 104664750 T^{3} + 173062868364 T^{4} - 104664750 p^{3} T^{5} - 31015 p^{6} T^{6} - 750 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( 1 - 692946 T^{2} + 197538719597 T^{4} - 70485548953384026 T^{6} + \)\(28\!\cdots\!96\)\( T^{8} - 70485548953384026 p^{6} T^{10} + 197538719597 p^{12} T^{12} - 692946 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( ( 1 - 161444 T^{2} + 37204239974 T^{4} - 161444 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 - 282 T - 631559 T^{2} + 18880182 T^{3} + 323368618692 T^{4} + 18880182 p^{3} T^{5} - 631559 p^{6} T^{6} - 282 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 79 | \( 1 - 1146210 T^{2} + 502439249741 T^{4} - 372728239473978570 T^{6} + \)\(27\!\cdots\!40\)\( T^{8} - 372728239473978570 p^{6} T^{10} + 502439249741 p^{12} T^{12} - 1146210 p^{18} T^{14} + p^{24} T^{16} \) |
| 83 | \( ( 1 + 923852 T^{2} + 770915307414 T^{4} + 923852 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - 562 T - 1018687 T^{2} + 42378734 T^{3} + 1061331470164 T^{4} + 42378734 p^{3} T^{5} - 1018687 p^{6} T^{6} - 562 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 468 T + 1472474 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
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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
−4.59805958467655717960830405096, −4.58526290361680825028720254376, −4.24059990520794168618150956974, −4.22713992518140597378459373486, −3.61694715909230718701801631884, −3.59870109460357444986273256747, −3.36234148172696656269738484420, −3.34494854307672403944560477556, −3.29737882533407458468948429742, −2.92289253161495456378496885518, −2.68667103044402886550062177208, −2.56182712563089115608487341114, −2.52573066679991170864817212482, −2.41518028756308107207684891979, −2.22724582464460601778755041222, −2.00802324430396377737735497636, −1.97659103046077127890209453405, −1.83444739439290877238167057743, −1.68693572826170205796715422778, −1.61839985966225762554680673649, −0.58167370657332786601467245714, −0.51794633726782619852695811252, −0.51534551262930475690305476368, −0.48547883088413933480355876629, −0.092321133334455867186479291243,
0.092321133334455867186479291243, 0.48547883088413933480355876629, 0.51534551262930475690305476368, 0.51794633726782619852695811252, 0.58167370657332786601467245714, 1.61839985966225762554680673649, 1.68693572826170205796715422778, 1.83444739439290877238167057743, 1.97659103046077127890209453405, 2.00802324430396377737735497636, 2.22724582464460601778755041222, 2.41518028756308107207684891979, 2.52573066679991170864817212482, 2.56182712563089115608487341114, 2.68667103044402886550062177208, 2.92289253161495456378496885518, 3.29737882533407458468948429742, 3.34494854307672403944560477556, 3.36234148172696656269738484420, 3.59870109460357444986273256747, 3.61694715909230718701801631884, 4.22713992518140597378459373486, 4.24059990520794168618150956974, 4.58526290361680825028720254376, 4.59805958467655717960830405096
Plot not available for L-functions of degree greater than 10.
