| L(s) = 1 | + 2·3-s + 9·5-s − 2·7-s − 3·9-s + 6·13-s + 18·15-s − 15·17-s − 16·19-s − 4·21-s + 9·23-s + 40·25-s − 14·27-s + 15·29-s − 18·35-s + 12·39-s + 3·41-s − 8·43-s − 27·45-s − 15·47-s − 9·49-s − 30·51-s − 9·53-s − 32·57-s + 9·59-s + 8·61-s + 6·63-s + 54·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 4.02·5-s − 0.755·7-s − 9-s + 1.66·13-s + 4.64·15-s − 3.63·17-s − 3.67·19-s − 0.872·21-s + 1.87·23-s + 8·25-s − 2.69·27-s + 2.78·29-s − 3.04·35-s + 1.92·39-s + 0.468·41-s − 1.21·43-s − 4.02·45-s − 2.18·47-s − 9/7·49-s − 4.20·51-s − 1.23·53-s − 4.23·57-s + 1.17·59-s + 1.02·61-s + 0.755·63-s + 6.69·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.852988394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.852988394\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $D_{4}$ | \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 174 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 15 T + 125 T^{2} + 750 T^{3} + 3486 T^{4} + 750 p T^{5} + 125 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 9 T + 77 T^{2} - 450 T^{3} + 2592 T^{4} - 450 p T^{5} + 77 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 15 T + 119 T^{2} - 720 T^{3} + 3870 T^{4} - 720 p T^{5} + 119 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 61 T^{2} + 2184 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 97 T^{2} + 5016 T^{4} - 97 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 3 T - T^{2} + 216 T^{3} - 1950 T^{4} + 216 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T - 5 T^{2} - 136 T^{3} + 160 T^{4} - 136 p T^{5} - 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 15 T + 185 T^{2} + 1650 T^{3} + 13416 T^{4} + 1650 p T^{5} + 185 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 9 T - 37 T^{2} + 108 T^{3} + 6822 T^{4} + 108 p T^{5} - 37 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 9 T - 49 T^{2} - 108 T^{3} + 8640 T^{4} - 108 p T^{5} - 49 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 41 T^{2} + 136 T^{3} + 3400 T^{4} + 136 p T^{5} - 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 3 T - 61 T^{2} - 216 T^{3} - 780 T^{4} - 216 p T^{5} - 61 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 16 T + 79 T^{2} + 496 T^{3} + 7312 T^{4} + 496 p T^{5} + 79 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 + 59 T^{2} - 2760 T^{4} + 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 28974 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 9 T - 109 T^{2} + 108 T^{3} + 20970 T^{4} + 108 p T^{5} - 109 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 9 T + 5 T^{2} - 198 T^{3} - 6306 T^{4} - 198 p T^{5} + 5 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.90047379003790784134866084390, −6.81340666805376471071190719917, −6.64348530397191925355040092686, −6.44412889087562354106942752003, −6.42577472185693613763910551368, −6.21790578957733815898589154544, −5.88824628047210110311841928379, −5.79633824400440947176760423398, −5.65081050990467645985753161368, −5.01251964626933171026057653204, −4.97648360924277591465545911459, −4.80990970981863814103446228049, −4.33532224219380630077427650815, −4.21245870042591326229214159312, −3.99403667326362975474386915649, −3.16041211862848878271416056519, −3.14558744002632417995563787656, −3.08612136267880570235649291592, −2.47180183643775377772621288990, −2.33090277534842738108762076190, −2.28767268263635189978519518316, −1.93177544090721717068697304151, −1.59009123824527270814174716050, −1.38869151938880182520436965990, −0.33659121273031930077034355304,
0.33659121273031930077034355304, 1.38869151938880182520436965990, 1.59009123824527270814174716050, 1.93177544090721717068697304151, 2.28767268263635189978519518316, 2.33090277534842738108762076190, 2.47180183643775377772621288990, 3.08612136267880570235649291592, 3.14558744002632417995563787656, 3.16041211862848878271416056519, 3.99403667326362975474386915649, 4.21245870042591326229214159312, 4.33532224219380630077427650815, 4.80990970981863814103446228049, 4.97648360924277591465545911459, 5.01251964626933171026057653204, 5.65081050990467645985753161368, 5.79633824400440947176760423398, 5.88824628047210110311841928379, 6.21790578957733815898589154544, 6.42577472185693613763910551368, 6.44412889087562354106942752003, 6.64348530397191925355040092686, 6.81340666805376471071190719917, 6.90047379003790784134866084390
