| L(s) = 1 | − 2·3-s + 4·5-s + 2·7-s + 5·9-s + 2·11-s − 4·13-s − 8·15-s + 4·17-s − 4·19-s − 4·21-s − 4·23-s + 12·25-s − 10·27-s − 12·29-s − 8·31-s − 4·33-s + 8·35-s + 16·37-s + 8·39-s − 16·41-s + 20·45-s − 4·47-s + 7·49-s − 8·51-s + 4·53-s + 8·55-s + 8·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 0.755·7-s + 5/3·9-s + 0.603·11-s − 1.10·13-s − 2.06·15-s + 0.970·17-s − 0.917·19-s − 0.872·21-s − 0.834·23-s + 12/5·25-s − 1.92·27-s − 2.22·29-s − 1.43·31-s − 0.696·33-s + 1.35·35-s + 2.63·37-s + 1.28·39-s − 2.49·41-s + 2.98·45-s − 0.583·47-s + 49-s − 1.12·51-s + 0.549·53-s + 1.07·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05672830905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05672830905\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4\times C_2$ | \( 1 - 4 T + 10 T^{2} + 112 T^{3} - 525 T^{4} + 112 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 24 T^{2} + 8 T^{3} + 935 T^{4} + 8 p T^{5} - 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T - 16 T^{2} - 56 T^{3} + 127 T^{4} - 56 p T^{5} - 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 16 T + 120 T^{2} - 992 T^{3} + 7655 T^{4} - 992 p T^{5} + 120 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 96 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 10 T^{2} - 272 T^{3} - 2285 T^{4} - 272 p T^{5} - 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} + 376 T^{3} - 3513 T^{4} + 376 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 14 T + 31 T^{2} + 658 T^{3} + 12180 T^{4} + 658 p T^{5} + 31 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 129 T^{2} + 1314 T^{3} + 14540 T^{4} + 1314 p T^{5} + 129 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14 T + 31 T^{2} - 434 T^{3} + 9604 T^{4} - 434 p T^{5} + 31 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 48 T^{2} - 992 T^{3} + 19247 T^{4} - 992 p T^{5} + 48 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 18 T + 103 T^{2} + 1134 T^{3} + 17004 T^{4} + 1134 p T^{5} + 103 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T - 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8 T - 58 T^{2} - 448 T^{3} + 1267 T^{4} - 448 p T^{5} - 58 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2 T + 187 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.80526593034520889293757107562, −6.65912576543505764352879724812, −6.57415811418307677688208446974, −6.18855204999598024099276178240, −5.90937410513882109939019523014, −5.90082833782051260858773696020, −5.59671282808120311801566003721, −5.54039280383235118055395296516, −5.08140903004630123525536019630, −5.02463484823249842840674078748, −4.83677143227443591637514099787, −4.50281456137945828038769451159, −4.36505882253948151849299092970, −3.89926753677861942613400968106, −3.80502141466561097451213847574, −3.43582504747756555725779251957, −3.35806701696570337299925675011, −2.50964540095253481216939716324, −2.47584334545804387681865554557, −2.30616989307676701334637295369, −1.89662910988810988491518488172, −1.48669507545229224292878013407, −1.28972389525511750117094385503, −1.23187416074648973047204664094, −0.04769916397000864123673790202,
0.04769916397000864123673790202, 1.23187416074648973047204664094, 1.28972389525511750117094385503, 1.48669507545229224292878013407, 1.89662910988810988491518488172, 2.30616989307676701334637295369, 2.47584334545804387681865554557, 2.50964540095253481216939716324, 3.35806701696570337299925675011, 3.43582504747756555725779251957, 3.80502141466561097451213847574, 3.89926753677861942613400968106, 4.36505882253948151849299092970, 4.50281456137945828038769451159, 4.83677143227443591637514099787, 5.02463484823249842840674078748, 5.08140903004630123525536019630, 5.54039280383235118055395296516, 5.59671282808120311801566003721, 5.90082833782051260858773696020, 5.90937410513882109939019523014, 6.18855204999598024099276178240, 6.57415811418307677688208446974, 6.65912576543505764352879724812, 6.80526593034520889293757107562
