| L(s) = 1 | + 4·3-s + 4·5-s − 4·7-s + 6·9-s + 4·13-s + 16·15-s − 4·17-s − 16·21-s − 12·23-s + 2·25-s − 4·27-s + 8·29-s − 16·35-s − 12·37-s + 16·39-s + 4·43-s + 24·45-s − 20·47-s + 8·49-s − 16·51-s + 12·53-s + 16·59-s + 24·61-s − 24·63-s + 16·65-s + 12·67-s − 48·69-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.78·5-s − 1.51·7-s + 2·9-s + 1.10·13-s + 4.13·15-s − 0.970·17-s − 3.49·21-s − 2.50·23-s + 2/5·25-s − 0.769·27-s + 1.48·29-s − 2.70·35-s − 1.97·37-s + 2.56·39-s + 0.609·43-s + 3.57·45-s − 2.91·47-s + 8/7·49-s − 2.24·51-s + 1.64·53-s + 2.08·59-s + 3.07·61-s − 3.02·63-s + 1.98·65-s + 1.46·67-s − 5.77·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.438398701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.438398701\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 20 T^{3} + 46 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 4 T^{3} - 194 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 444 T^{3} + 2542 T^{4} + 444 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 468 T^{3} + 3038 T^{4} + 468 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 164 T^{3} + 3358 T^{4} - 164 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1860 T^{3} + 15182 T^{4} + 1860 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} + 180 T^{3} - 6274 T^{4} + 180 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 13286 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 44 T^{3} - 3602 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 196 T^{3} + 3646 T^{4} - 196 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.808129824008952437785844539875, −9.789972872220909503903022876028, −9.453221157055196676060724624385, −8.965225070954575663481854249117, −8.961190850250557643550041719968, −8.413554427737703713322708483129, −8.314032440063298417707908197741, −8.211790100487212882458537813355, −8.047870011095552443531621300091, −7.11596850478304211230901869815, −7.05663812198012161711907642961, −6.72945867873405848484556620415, −6.53477497105316379026228552779, −5.91111119955846777214968353710, −5.77062408929591675194553800210, −5.71656119538759605097942690165, −5.12678474282479562964715783389, −4.45811822690770584955440712902, −3.94281134066803168869233712163, −3.60671944413370315456148518299, −3.57451418291999757664576101392, −2.90112192122699118115711668834, −2.40472112272085977316862871453, −2.13978100346457298606810073123, −1.81271643370932785045831121720,
1.81271643370932785045831121720, 2.13978100346457298606810073123, 2.40472112272085977316862871453, 2.90112192122699118115711668834, 3.57451418291999757664576101392, 3.60671944413370315456148518299, 3.94281134066803168869233712163, 4.45811822690770584955440712902, 5.12678474282479562964715783389, 5.71656119538759605097942690165, 5.77062408929591675194553800210, 5.91111119955846777214968353710, 6.53477497105316379026228552779, 6.72945867873405848484556620415, 7.05663812198012161711907642961, 7.11596850478304211230901869815, 8.047870011095552443531621300091, 8.211790100487212882458537813355, 8.314032440063298417707908197741, 8.413554427737703713322708483129, 8.961190850250557643550041719968, 8.965225070954575663481854249117, 9.453221157055196676060724624385, 9.789972872220909503903022876028, 9.808129824008952437785844539875
