| L(s) = 1 | − 8·7-s − 18·9-s + 56·17-s + 112·23-s − 12·25-s + 552·31-s − 1.17e3·41-s + 1.42e3·47-s − 852·49-s + 144·63-s − 432·71-s − 728·73-s + 2.15e3·79-s + 243·81-s − 4.61e3·89-s − 3.80e3·97-s − 1.40e3·113-s − 448·119-s + 2.60e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.00e3·153-s + 157-s + ⋯ |
| L(s) = 1 | − 0.431·7-s − 2/3·9-s + 0.798·17-s + 1.01·23-s − 0.0959·25-s + 3.19·31-s − 4.47·41-s + 4.41·47-s − 2.48·49-s + 0.287·63-s − 0.722·71-s − 1.16·73-s + 3.06·79-s + 1/3·81-s − 5.49·89-s − 3.97·97-s − 1.16·113-s − 0.345·119-s + 1.95·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.532·153-s + 0.000508·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.05432215380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05432215380\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 12 T^{2} + 15926 T^{4} + 12 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 4 T + 450 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2604 T^{2} + 3702326 T^{4} - 2604 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 4390 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 28 T + 1382 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 19244 T^{2} + 174632406 T^{4} - 19244 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 56 T + 16478 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6068 T^{2} + 357799638 T^{4} - 6068 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 276 T + 66866 T^{2} - 276 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 32492 T^{2} + 5072441334 T^{4} - 32492 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 588 T + 223318 T^{2} + 588 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 81100 T^{2} + 13216498038 T^{4} - 81100 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 712 T + 333422 T^{2} - 712 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 14380 T^{2} + 44087902518 T^{4} + 14380 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 401388 T^{2} + 84923349878 T^{4} - 401388 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 481292 T^{2} + 127804965078 T^{4} - 481292 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 346036 T^{2} + 144908686422 T^{4} + 346036 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 216 T + 718846 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 364 T + 162198 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 1076 T + 1221522 T^{2} - 1076 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 232076 T^{2} + 279958929942 T^{4} - 232076 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 2308 T + 2645654 T^{2} + 2308 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 1900 T + 2631846 T^{2} + 1900 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.84594099855234328358537779127, −6.76809799488619701190610261779, −6.62500695517754024197282064812, −6.49799699986195001134482391394, −5.93333069154456838245464918016, −5.77576702197598703407026996751, −5.74114127389941636193470445324, −5.24652221683574679498534099653, −5.16874392015044275384132208704, −4.85187538445221409939812713628, −4.79043410669180017061934448583, −4.24983372362757565810852829301, −3.98294204215354655278275429942, −3.78872882975979297200994416032, −3.72204717212705179367673327804, −2.84876835047285866961288060134, −2.83339057551255209430523845412, −2.83090338989683317261721986290, −2.81453199426639482877541690939, −2.01026714833337496829604568514, −1.47200820639227776177835275580, −1.44967715275262932548724120755, −1.06475525630631999841239480939, −0.59230776421474514896159894902, −0.03391869382334538309420541579,
0.03391869382334538309420541579, 0.59230776421474514896159894902, 1.06475525630631999841239480939, 1.44967715275262932548724120755, 1.47200820639227776177835275580, 2.01026714833337496829604568514, 2.81453199426639482877541690939, 2.83090338989683317261721986290, 2.83339057551255209430523845412, 2.84876835047285866961288060134, 3.72204717212705179367673327804, 3.78872882975979297200994416032, 3.98294204215354655278275429942, 4.24983372362757565810852829301, 4.79043410669180017061934448583, 4.85187538445221409939812713628, 5.16874392015044275384132208704, 5.24652221683574679498534099653, 5.74114127389941636193470445324, 5.77576702197598703407026996751, 5.93333069154456838245464918016, 6.49799699986195001134482391394, 6.62500695517754024197282064812, 6.76809799488619701190610261779, 6.84594099855234328358537779127
