| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s − 8·5-s + 4·6-s + 2·8-s + 2·9-s − 16·10-s − 4·11-s + 4·12-s − 8·13-s − 16·15-s + 3·16-s − 4·17-s + 4·18-s + 4·19-s − 16·20-s − 8·22-s − 10·23-s + 4·24-s + 38·25-s − 16·26-s + 4·27-s − 32·30-s + 8·32-s − 8·33-s − 8·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s − 3.57·5-s + 1.63·6-s + 0.707·8-s + 2/3·9-s − 5.05·10-s − 1.20·11-s + 1.15·12-s − 2.21·13-s − 4.13·15-s + 3/4·16-s − 0.970·17-s + 0.942·18-s + 0.917·19-s − 3.57·20-s − 1.70·22-s − 2.08·23-s + 0.816·24-s + 38/5·25-s − 3.13·26-s + 0.769·27-s − 5.84·30-s + 1.41·32-s − 1.39·33-s − 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9081410578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9081410578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + p T^{2} - p T^{3} + T^{4} - p^{2} T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 4 T^{3} + 7 T^{4} - 4 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 52 T^{3} + 322 T^{4} + 52 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 10 T + 50 T^{2} + 220 T^{3} + 967 T^{4} + 220 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2310 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 238 T^{4} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 60 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 492 T^{3} + 3326 T^{4} - 492 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 170 T^{2} + 13467 T^{4} - 170 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14 T + 98 T^{2} - 756 T^{3} + 5663 T^{4} - 756 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2904 T^{3} + 26978 T^{4} + 2904 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 29082 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 140 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.691759881294683334297403684397, −8.469868201221975543268290845509, −8.080342089127281898290420245919, −7.935978167230126076750802931832, −7.70904115632614330982076640713, −7.66823542760922259889363221189, −7.46632750674137282258092335135, −6.93716330113541154079966650838, −6.91055541017455351932030985046, −6.56615396166705293942743232542, −6.26557337736235824925591136867, −5.56379098918901488564576068286, −5.15217499296133679029097094006, −5.10115800297604151579559065258, −4.98204515741152713416999143793, −4.25927275097120126104691325460, −4.12242464033419056172811723613, −4.10693277585559435231597070057, −3.94561733153180903237663335026, −3.17855357053246649828521883377, −2.99675848133780814309802307991, −2.73251224295752982209780022436, −2.52601759322983666213438995356, −1.70239968184969128747197272934, −0.36313216493680953485108363607,
0.36313216493680953485108363607, 1.70239968184969128747197272934, 2.52601759322983666213438995356, 2.73251224295752982209780022436, 2.99675848133780814309802307991, 3.17855357053246649828521883377, 3.94561733153180903237663335026, 4.10693277585559435231597070057, 4.12242464033419056172811723613, 4.25927275097120126104691325460, 4.98204515741152713416999143793, 5.10115800297604151579559065258, 5.15217499296133679029097094006, 5.56379098918901488564576068286, 6.26557337736235824925591136867, 6.56615396166705293942743232542, 6.91055541017455351932030985046, 6.93716330113541154079966650838, 7.46632750674137282258092335135, 7.66823542760922259889363221189, 7.70904115632614330982076640713, 7.935978167230126076750802931832, 8.080342089127281898290420245919, 8.469868201221975543268290845509, 8.691759881294683334297403684397
