| L(s) = 1 | + 2-s − 2·3-s + 5-s − 2·6-s + 10·7-s − 5·8-s + 9-s + 10-s + 4·11-s − 8·13-s + 10·14-s − 2·15-s − 3·16-s + 11·17-s + 18-s − 7·19-s − 20·21-s + 4·22-s − 2·23-s + 10·24-s + 6·25-s − 8·26-s + 2·27-s − 20·29-s − 2·30-s + 31-s − 8·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 0.447·5-s − 0.816·6-s + 3.77·7-s − 1.76·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 2.21·13-s + 2.67·14-s − 0.516·15-s − 3/4·16-s + 2.66·17-s + 0.235·18-s − 1.60·19-s − 4.36·21-s + 0.852·22-s − 0.417·23-s + 2.04·24-s + 6/5·25-s − 1.56·26-s + 0.384·27-s − 3.71·29-s − 0.365·30-s + 0.179·31-s − 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.056004812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.056004812\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T + T^{2} + p^{2} T^{3} - 3 p T^{4} + p^{3} T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - T - p T^{2} + 4 T^{3} + 6 T^{4} + 4 p T^{5} - p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 11 T + 61 T^{2} - 286 T^{3} + 1254 T^{4} - 286 p T^{5} + 61 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 + 7 T + 3 T^{2} + 56 T^{3} + 824 T^{4} + 56 p T^{5} + 3 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - T - 23 T^{2} + 38 T^{3} - 416 T^{4} + 38 p T^{5} - 23 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 11 T + 21 T^{2} + 286 T^{3} + 4154 T^{4} + 286 p T^{5} + 21 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 13 T + 90 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + T - 55 T^{2} - 38 T^{3} + 880 T^{4} - 38 p T^{5} - 55 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 5 T + 19 T^{2} - 500 T^{3} - 4098 T^{4} - 500 p T^{5} + 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 10 T - 26 T^{2} + 80 T^{3} + 6495 T^{4} + 80 p T^{5} - 26 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 20 T + 183 T^{2} - 1660 T^{3} + 15800 T^{4} - 1660 p T^{5} + 183 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 20 T + 171 T^{2} + 1660 T^{3} + 17912 T^{4} + 1660 p T^{5} + 171 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 15 T + 15 T^{2} - 780 T^{3} + 20084 T^{4} - 780 p T^{5} + 15 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 18 T + 230 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.059227927735918481623161656023, −7.54572658791151997267091225868, −7.52638248709012517640663729645, −7.24454664749976156119705391788, −7.02369773270321192426659421296, −6.54605009165360557606150031686, −6.36541954870547248708575720442, −6.10599959857684886238452247235, −5.93529181365315807535493468796, −5.38720340945721854974327445068, −5.34091302821197766296043342192, −5.08570418477989617712130426177, −5.04431152127747262566345262315, −4.91837024067021680092175105891, −4.67512297750776724756958899021, −4.09460737494508445701393598870, −3.60921535758832814585326049452, −3.60664737231626261627441002031, −3.39228376010483010236199504431, −2.80406520882602336697866095711, −2.08539253522676151152463374339, −1.92713485276118189797013158108, −1.78568754583796365940005761725, −1.36782226818114186719182331256, −0.45533142305741113221043077737,
0.45533142305741113221043077737, 1.36782226818114186719182331256, 1.78568754583796365940005761725, 1.92713485276118189797013158108, 2.08539253522676151152463374339, 2.80406520882602336697866095711, 3.39228376010483010236199504431, 3.60664737231626261627441002031, 3.60921535758832814585326049452, 4.09460737494508445701393598870, 4.67512297750776724756958899021, 4.91837024067021680092175105891, 5.04431152127747262566345262315, 5.08570418477989617712130426177, 5.34091302821197766296043342192, 5.38720340945721854974327445068, 5.93529181365315807535493468796, 6.10599959857684886238452247235, 6.36541954870547248708575720442, 6.54605009165360557606150031686, 7.02369773270321192426659421296, 7.24454664749976156119705391788, 7.52638248709012517640663729645, 7.54572658791151997267091225868, 8.059227927735918481623161656023
