| L(s) = 1 | + 4·4-s − 4·5-s + 4·7-s + 7·16-s − 16·20-s + 10·25-s + 16·28-s − 16·35-s + 8·37-s + 8·43-s + 24·47-s − 2·49-s + 24·59-s + 8·64-s − 16·67-s − 16·79-s − 28·80-s − 24·83-s + 24·89-s + 40·100-s + 32·109-s + 28·112-s + 16·121-s − 20·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1.78·5-s + 1.51·7-s + 7/4·16-s − 3.57·20-s + 2·25-s + 3.02·28-s − 2.70·35-s + 1.31·37-s + 1.21·43-s + 3.50·47-s − 2/7·49-s + 3.12·59-s + 64-s − 1.95·67-s − 1.80·79-s − 3.13·80-s − 2.63·83-s + 2.54·89-s + 4·100-s + 3.06·109-s + 2.64·112-s + 1.45·121-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.095900507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.095900507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 114 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 486 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2934 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 5826 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1206 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 78 p T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 25110 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 15606 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.529520374171114882506815604666, −7.942762985979819311952952182424, −7.941245228743110013209012295908, −7.54843483348540299013991440282, −7.51398810241649554935762631301, −7.38450281613412684332292429467, −7.00660177259172093209196409977, −6.89055431914241155868229891963, −6.42599872617357549692423868531, −6.19491046902529440168830119634, −5.95927675528890213325119011227, −5.54077156044788334666290295041, −5.39064607497294255953299002215, −4.99995413466858911931645253680, −4.53115895104720968665450312972, −4.35825056307535366849534896428, −4.09896517338678989302447389888, −3.87004547748875191951105188808, −3.32965442936925613306627543020, −3.02764358493420422960875468139, −2.50460288368402098040666663796, −2.41677867244670426775587416637, −1.97407916811687505288819705787, −1.30303491816795261239176676087, −0.854850235103496646194412884209,
0.854850235103496646194412884209, 1.30303491816795261239176676087, 1.97407916811687505288819705787, 2.41677867244670426775587416637, 2.50460288368402098040666663796, 3.02764358493420422960875468139, 3.32965442936925613306627543020, 3.87004547748875191951105188808, 4.09896517338678989302447389888, 4.35825056307535366849534896428, 4.53115895104720968665450312972, 4.99995413466858911931645253680, 5.39064607497294255953299002215, 5.54077156044788334666290295041, 5.95927675528890213325119011227, 6.19491046902529440168830119634, 6.42599872617357549692423868531, 6.89055431914241155868229891963, 7.00660177259172093209196409977, 7.38450281613412684332292429467, 7.51398810241649554935762631301, 7.54843483348540299013991440282, 7.941245228743110013209012295908, 7.942762985979819311952952182424, 8.529520374171114882506815604666
