| L(s) = 1 | − 3·3-s + 9·5-s + 19·7-s − 21·9-s − 24·11-s − 61·13-s − 27·15-s + 6·17-s − 266·19-s − 57·21-s + 69·23-s + 34·25-s + 72·27-s − 237·29-s + 211·31-s + 72·33-s + 171·35-s + 524·37-s + 183·39-s − 468·41-s − 86·43-s − 189·45-s + 483·47-s + 540·49-s − 18·51-s + 300·53-s − 216·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.804·5-s + 1.02·7-s − 7/9·9-s − 0.657·11-s − 1.30·13-s − 0.464·15-s + 0.0856·17-s − 3.21·19-s − 0.592·21-s + 0.625·23-s + 0.271·25-s + 0.513·27-s − 1.51·29-s + 1.22·31-s + 0.379·33-s + 0.825·35-s + 2.32·37-s + 0.751·39-s − 1.78·41-s − 0.304·43-s − 0.626·45-s + 1.49·47-s + 1.57·49-s − 0.0494·51-s + 0.777·53-s − 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8548842012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8548842012\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T + 10 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 9 T + 47 T^{2} + 1944 T^{3} - 24594 T^{4} + 1944 p^{3} T^{5} + 47 p^{6} T^{6} - 9 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 19 T - 179 T^{2} + 2774 T^{3} + 50128 T^{4} + 2774 p^{3} T^{5} - 179 p^{6} T^{6} - 19 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 24 T - 1285 T^{2} - 19224 T^{3} + 925104 T^{4} - 19224 p^{3} T^{5} - 1285 p^{6} T^{6} + 24 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 61 T - 1367 T^{2} + 42334 T^{3} + 12885898 T^{4} + 42334 p^{3} T^{5} - 1367 p^{6} T^{6} + 61 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 9592 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 7 p T + 12234 T^{2} + 7 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 p T - 20527 T^{2} - 2862 p T^{3} + 433519968 T^{4} - 2862 p^{4} T^{5} - 20527 p^{6} T^{6} - 3 p^{10} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 237 T + 4925 T^{2} + 584442 T^{3} + 661218474 T^{4} + 584442 p^{3} T^{5} + 4925 p^{6} T^{6} + 237 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 211 T - 24065 T^{2} - 1899844 T^{3} + 2490210604 T^{4} - 1899844 p^{3} T^{5} - 24065 p^{6} T^{6} - 211 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 262 T + 72162 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 468 T + 27371 T^{2} + 25183548 T^{3} + 16885415064 T^{4} + 25183548 p^{3} T^{5} + 27371 p^{6} T^{6} + 468 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 p T - 17387 T^{2} - 268462 p T^{3} - 6295199732 T^{4} - 268462 p^{4} T^{5} - 17387 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 483 T + 20477 T^{2} - 2495178 T^{3} + 10288968168 T^{4} - 2495178 p^{3} T^{5} + 20477 p^{6} T^{6} - 483 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 150 T + 257074 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 168 T - 388645 T^{2} - 1026648 T^{3} + 125802612624 T^{4} - 1026648 p^{3} T^{5} - 388645 p^{6} T^{6} - 168 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 1049 T + 424495 T^{2} - 232819256 T^{3} + 155558427094 T^{4} - 232819256 p^{3} T^{5} + 424495 p^{6} T^{6} - 1049 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 1166 T + 452161 T^{2} + 356643254 T^{3} + 324003162628 T^{4} + 356643254 p^{3} T^{5} + 452161 p^{6} T^{6} + 1166 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 312 T + 498238 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 311 T + 698028 T^{2} + 311 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 349 T - 854801 T^{2} + 3307124 T^{3} + 650611367644 T^{4} + 3307124 p^{3} T^{5} - 854801 p^{6} T^{6} - 349 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1221 T + 78743 T^{2} - 327867804 T^{3} + 814636885368 T^{4} - 327867804 p^{3} T^{5} + 78743 p^{6} T^{6} - 1221 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 492 T + 1092454 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 128 T - 1698713 T^{2} + 14111872 T^{3} + 2093632480048 T^{4} + 14111872 p^{3} T^{5} - 1698713 p^{6} T^{6} - 128 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−9.108506839391095159173752686770, −9.046248919664430857333052269333, −8.607727120676038461233447069867, −8.162767386362665182399056347473, −8.097750545001522957213848011347, −7.68746142352250464032397680888, −7.67420606599907610094816476089, −6.88841161212008414423500520905, −6.85759439982233999872364994338, −6.46813269483645558087273651634, −6.32565503876575867342924246389, −5.72259319766104487392974366358, −5.52311506444878130495264468891, −5.31197205094456153531261928730, −5.16418555081411662806657424604, −4.39789281816210859011363966552, −4.32322668023791374867089543492, −4.19023838953398708305929820665, −3.43451287780212976791634826380, −2.66126636461636690060314157570, −2.54997433419926631153620631299, −2.20855031699205121941629666626, −1.78417780228255051950180760606, −0.953611218582310441933731134937, −0.24693209726802240403319622104,
0.24693209726802240403319622104, 0.953611218582310441933731134937, 1.78417780228255051950180760606, 2.20855031699205121941629666626, 2.54997433419926631153620631299, 2.66126636461636690060314157570, 3.43451287780212976791634826380, 4.19023838953398708305929820665, 4.32322668023791374867089543492, 4.39789281816210859011363966552, 5.16418555081411662806657424604, 5.31197205094456153531261928730, 5.52311506444878130495264468891, 5.72259319766104487392974366358, 6.32565503876575867342924246389, 6.46813269483645558087273651634, 6.85759439982233999872364994338, 6.88841161212008414423500520905, 7.67420606599907610094816476089, 7.68746142352250464032397680888, 8.097750545001522957213848011347, 8.162767386362665182399056347473, 8.607727120676038461233447069867, 9.046248919664430857333052269333, 9.108506839391095159173752686770
