| L(s) = 1 | − 2·2-s + 2·4-s − 6·5-s − 4·8-s + 12·10-s − 6·13-s + 8·16-s − 12·20-s + 11·25-s + 12·26-s + 10·29-s − 8·32-s + 24·40-s + 30·41-s − 2·49-s − 22·50-s − 12·52-s − 16·53-s − 20·58-s + 18·61-s + 8·64-s + 36·65-s − 48·80-s − 60·82-s + 18·97-s + 4·98-s + 22·100-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 2.68·5-s − 1.41·8-s + 3.79·10-s − 1.66·13-s + 2·16-s − 2.68·20-s + 11/5·25-s + 2.35·26-s + 1.85·29-s − 1.41·32-s + 3.79·40-s + 4.68·41-s − 2/7·49-s − 3.11·50-s − 1.66·52-s − 2.19·53-s − 2.62·58-s + 2.30·61-s + 64-s + 4.46·65-s − 5.36·80-s − 6.62·82-s + 1.82·97-s + 0.404·98-s + 11/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3760186744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3760186744\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 21 T^{2} - 88 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 + 59 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 15 T + 116 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 67 T^{2} + 2280 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - 43 T^{2} - 1632 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 91 T^{2} + 3792 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 149 T^{2} + 15960 T^{4} + 149 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^3$ | \( 1 - 163 T^{2} + 19680 T^{4} - 163 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
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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
−7.53519115182901354943397194829, −7.41000380764220006565734402571, −7.15113471480405765249119608047, −7.13675092744330821070774135149, −6.38940493554191767021458088283, −6.36287540646109281927825085395, −6.15854377186646824293055521866, −5.97518713117145819464418938427, −5.72325639578838715584338211905, −5.19044634702820855042486595199, −4.87279657156461920240910539850, −4.77495606456586541525031987290, −4.45931133321390134214986435065, −4.39764508340137968662025273542, −3.84586497068514012862099283838, −3.73843150086036358401906539328, −3.36210505924807439539173357009, −3.26917324115554641724214916692, −2.72892097497309580346862467534, −2.55522509767766928135146504693, −2.23858337626806081420893150602, −1.85442630039743405572220415730, −0.906404950604827799280188765987, −0.73911426873657444621242409896, −0.38094486110590302123934781681,
0.38094486110590302123934781681, 0.73911426873657444621242409896, 0.906404950604827799280188765987, 1.85442630039743405572220415730, 2.23858337626806081420893150602, 2.55522509767766928135146504693, 2.72892097497309580346862467534, 3.26917324115554641724214916692, 3.36210505924807439539173357009, 3.73843150086036358401906539328, 3.84586497068514012862099283838, 4.39764508340137968662025273542, 4.45931133321390134214986435065, 4.77495606456586541525031987290, 4.87279657156461920240910539850, 5.19044634702820855042486595199, 5.72325639578838715584338211905, 5.97518713117145819464418938427, 6.15854377186646824293055521866, 6.36287540646109281927825085395, 6.38940493554191767021458088283, 7.13675092744330821070774135149, 7.15113471480405765249119608047, 7.41000380764220006565734402571, 7.53519115182901354943397194829
