| L(s) = 1 | + 676·7-s − 3.44e3·13-s + 3.66e3·19-s + 3.99e4·25-s + 1.53e5·31-s + 1.28e5·37-s − 1.26e5·43-s + 5.41e4·49-s + 1.67e5·61-s − 8·67-s − 1.88e6·73-s − 1.18e6·79-s − 2.33e6·91-s + 9.75e5·97-s + 2.91e6·103-s + 1.79e6·109-s + 2.37e5·121-s + 127-s + 131-s + 2.47e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.97·7-s − 1.56·13-s + 0.534·19-s + 2.55·25-s + 5.14·31-s + 2.54·37-s − 1.58·43-s + 0.460·49-s + 0.738·61-s − 2.65e − 5·67-s − 4.83·73-s − 2.41·79-s − 3.09·91-s + 1.06·97-s + 2.66·103-s + 1.38·109-s + 0.133·121-s + 1.05·133-s − 2.26·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(8.167645164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.167645164\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 39946 T^{2} + 1326939 p^{4} T^{4} - 39946 p^{12} T^{6} + p^{24} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 338 T + 144303 T^{2} - 338 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 237010 T^{2} + 2525538486003 T^{4} - 237010 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 1724 T + 9918438 T^{2} + 1724 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 89603068 T^{2} + 3171430277926278 T^{4} - 89603068 p^{12} T^{6} + p^{24} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 1832 T - 12669582 T^{2} - 1832 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 163687972 T^{2} + 40032274676179974 T^{4} - 163687972 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1305806620 T^{2} + 920788083150632358 T^{4} - 1305806620 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 76622 T + 3096283983 T^{2} - 76622 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 64480 T + 6063270018 T^{2} - 64480 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 16738605220 T^{2} + \)\(11\!\cdots\!78\)\( T^{4} - 16738605220 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 63004 T + 13577236998 T^{2} + 63004 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 38620471036 T^{2} + \)\(60\!\cdots\!30\)\( T^{4} - 38620471036 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 36815214586 T^{2} + \)\(92\!\cdots\!55\)\( T^{4} - 36815214586 p^{12} T^{6} + p^{24} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 32474880220 T^{2} + \)\(21\!\cdots\!78\)\( T^{4} - 32474880220 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 83848 T + 100016130498 T^{2} - 83848 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 4 T + 35727479718 T^{2} + 4 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 40430249500 T^{2} - \)\(15\!\cdots\!82\)\( T^{4} - 40430249500 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 940838 T + 438394346043 T^{2} + 940838 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 594364 T + 526738516422 T^{2} + 594364 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 477023926306 T^{2} + \)\(10\!\cdots\!95\)\( T^{4} - 477023926306 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 300686549980 T^{2} + \)\(33\!\cdots\!18\)\( T^{4} - 300686549980 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 487606 T + 985315056603 T^{2} - 487606 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−7.01612488892556659806882499981, −6.93407639307280333940833026428, −6.49546029468015725344314494519, −6.37493371466697087136046172228, −5.91390956222213169939680180956, −5.91294020039140036813248847552, −5.55048410933135486111023407236, −5.03259327041158062224150919591, −4.84024744551955935721030479647, −4.69567970589258525726233039317, −4.56918282796942505516491172387, −4.38707955461575590631275869966, −4.38026533518429285375431446232, −3.44435973473905694953211807832, −3.31715905810621791688928784758, −2.85462195149071528559291401311, −2.77497941067313750670921538045, −2.55486496616125158953299750413, −2.24932033957582826849684122083, −1.74387422980680102016981780176, −1.38177742264375748190054313765, −1.23107245938302446257215076319, −0.862966632279656639426129720538, −0.71499911425437439297349103663, −0.26079703321869842586272472999,
0.26079703321869842586272472999, 0.71499911425437439297349103663, 0.862966632279656639426129720538, 1.23107245938302446257215076319, 1.38177742264375748190054313765, 1.74387422980680102016981780176, 2.24932033957582826849684122083, 2.55486496616125158953299750413, 2.77497941067313750670921538045, 2.85462195149071528559291401311, 3.31715905810621791688928784758, 3.44435973473905694953211807832, 4.38026533518429285375431446232, 4.38707955461575590631275869966, 4.56918282796942505516491172387, 4.69567970589258525726233039317, 4.84024744551955935721030479647, 5.03259327041158062224150919591, 5.55048410933135486111023407236, 5.91294020039140036813248847552, 5.91390956222213169939680180956, 6.37493371466697087136046172228, 6.49546029468015725344314494519, 6.93407639307280333940833026428, 7.01612488892556659806882499981
