| L(s) = 1 | + 2·2-s + 4-s − 2·5-s − 2·8-s − 4·10-s + 4·11-s − 8·13-s − 4·16-s − 4·17-s − 2·19-s − 2·20-s + 8·22-s + 6·23-s + 25-s − 16·26-s − 4·31-s − 2·32-s − 8·34-s − 4·37-s − 4·38-s + 4·40-s − 16·41-s + 24·43-s + 4·44-s + 12·46-s + 2·47-s + 14·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.894·5-s − 0.707·8-s − 1.26·10-s + 1.20·11-s − 2.21·13-s − 16-s − 0.970·17-s − 0.458·19-s − 0.447·20-s + 1.70·22-s + 1.25·23-s + 1/5·25-s − 3.13·26-s − 0.718·31-s − 0.353·32-s − 1.37·34-s − 0.657·37-s − 0.648·38-s + 0.632·40-s − 2.49·41-s + 3.65·43-s + 0.603·44-s + 1.76·46-s + 0.291·47-s + 2·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1681135727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1681135727\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 4 T - 3 T^{2} + 12 T^{3} + 64 T^{4} + 12 p T^{5} - 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} - 96 T^{3} - 461 T^{4} - 96 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 7 T^{2} - 54 T^{3} - 316 T^{4} - 54 p T^{5} - 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + T^{2} - 236 T^{3} - 1736 T^{4} - 236 p T^{5} + p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 91 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2 T + 21 T^{2} + 222 T^{3} - 2108 T^{4} + 222 p T^{5} + 21 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 10 T - 3 T^{2} + 30 T^{3} + 2500 T^{4} + 30 p T^{5} - 3 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T - 42 T^{2} - 96 T^{3} + 3979 T^{4} - 96 p T^{5} - 42 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 14 T^{2} + 96 T^{3} + 4395 T^{4} + 96 p T^{5} + 14 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 20 T + 194 T^{2} + 1440 T^{3} + 11147 T^{4} + 1440 p T^{5} + 194 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 12 T - 22 T^{2} + 96 T^{3} + 9939 T^{4} + 96 p T^{5} - 22 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 20 T + 238 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 8 T - 18 T^{2} - 768 T^{3} - 6893 T^{4} - 768 p T^{5} - 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
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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
−7.30967207704915222522038721120, −7.21327315330696162195216130846, −7.19677243180717480241731820263, −7.02404575623162728674459159839, −6.73968559820279947960253948716, −6.55641637862877829259353852915, −5.96902246016663420650824861868, −5.86809985297340354358007695655, −5.68443977604347836822360161270, −5.36176574195248816607668149407, −5.27201720699395148238851637895, −4.69911878429466426706604232578, −4.66913807214883490127455936487, −4.33138524683159453624806246133, −4.22120841438400863230902083749, −4.03539013202393980848623522385, −3.74669549203566876781595131752, −3.19674404407841627328342028108, −3.18727320140620887071096029570, −2.62691098707365688650859915755, −2.39211587519885662535693945185, −2.32248839518092677922451884960, −1.41773572250453284488507145049, −1.23421690309365059776694844834, −0.094428370817041684284242128261,
0.094428370817041684284242128261, 1.23421690309365059776694844834, 1.41773572250453284488507145049, 2.32248839518092677922451884960, 2.39211587519885662535693945185, 2.62691098707365688650859915755, 3.18727320140620887071096029570, 3.19674404407841627328342028108, 3.74669549203566876781595131752, 4.03539013202393980848623522385, 4.22120841438400863230902083749, 4.33138524683159453624806246133, 4.66913807214883490127455936487, 4.69911878429466426706604232578, 5.27201720699395148238851637895, 5.36176574195248816607668149407, 5.68443977604347836822360161270, 5.86809985297340354358007695655, 5.96902246016663420650824861868, 6.55641637862877829259353852915, 6.73968559820279947960253948716, 7.02404575623162728674459159839, 7.19677243180717480241731820263, 7.21327315330696162195216130846, 7.30967207704915222522038721120
