L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 2·5-s − 4·6-s − 2·7-s + 2·8-s + 9-s + 4·10-s + 11-s + 2·12-s − 2·13-s + 4·14-s − 4·15-s − 4·16-s + 8·17-s − 2·18-s + 19-s − 2·20-s − 4·21-s − 2·22-s − 6·23-s + 4·24-s − 9·25-s + 4·26-s − 2·27-s − 2·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s − 0.755·7-s + 0.707·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s + 0.577·12-s − 0.554·13-s + 1.06·14-s − 1.03·15-s − 16-s + 1.94·17-s − 0.471·18-s + 0.229·19-s − 0.447·20-s − 0.872·21-s − 0.426·22-s − 1.25·23-s + 0.816·24-s − 9/5·25-s + 0.784·26-s − 0.384·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6586901891\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6586901891\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
good | 5 | $D_{4}$ | \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - T - 17 T^{2} + 4 T^{3} + 192 T^{4} + 4 p T^{5} - 17 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T + 31 T^{2} + 8 T^{3} - 288 T^{4} + 8 p T^{5} + 31 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - T - 33 T^{2} + 4 T^{3} + 776 T^{4} + 4 p T^{5} - 33 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 2 T^{2} - 48 T^{3} + 87 T^{4} - 48 p T^{5} - 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - p T^{2} + 52 T^{3} + 720 T^{4} + 52 p T^{5} - p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 9 T - 9 T^{2} - 144 T^{3} + 3734 T^{4} - 144 p T^{5} - 9 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T + 13 T^{2} + 354 T^{3} - 2628 T^{4} + 354 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 2 T - 98 T^{2} - 32 T^{3} + 6687 T^{4} - 32 p T^{5} - 98 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 27 T^{2} + 354 T^{3} - 1948 T^{4} + 354 p T^{5} - 27 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 10 T - 42 T^{2} + 80 T^{3} + 8975 T^{4} + 80 p T^{5} - 42 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 2 T + 14 T^{2} - 304 T^{3} - 5225 T^{4} - 304 p T^{5} + 14 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 9 T + 162 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 9 T + 174 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 2 T + 150 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 11 T - 83 T^{2} + 286 T^{3} + 22926 T^{4} + 286 p T^{5} - 83 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 10 T - 102 T^{2} - 80 T^{3} + 21695 T^{4} - 80 p T^{5} - 102 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.986876294054291129943389105104, −7.73248284358295560912386242723, −7.39130873804695683583169536346, −7.25387520366286420394293603740, −7.18662181021587355110342057432, −6.81739302926739690352145119055, −6.36350501819475875124034433767, −6.10258884181386150685882078403, −5.84798205387652453239356846185, −5.64475336053364533715147800330, −5.57700340090638447950390809506, −5.07914427365449878916410289222, −4.55000271140865864625408407922, −4.39369288985163412354208430824, −4.10935545292385730249265189068, −3.98302415572056521904535188870, −3.53011459871439766322861903628, −3.46046007163665308964432715682, −2.81584964406598927557083718058, −2.68469348378206225240669971916, −2.55792539260442162263368390087, −1.73122446097542503798652451887, −1.62359781957480983878608191498, −0.939578901494111557970680690150, −0.37352115187422595505984512029,
0.37352115187422595505984512029, 0.939578901494111557970680690150, 1.62359781957480983878608191498, 1.73122446097542503798652451887, 2.55792539260442162263368390087, 2.68469348378206225240669971916, 2.81584964406598927557083718058, 3.46046007163665308964432715682, 3.53011459871439766322861903628, 3.98302415572056521904535188870, 4.10935545292385730249265189068, 4.39369288985163412354208430824, 4.55000271140865864625408407922, 5.07914427365449878916410289222, 5.57700340090638447950390809506, 5.64475336053364533715147800330, 5.84798205387652453239356846185, 6.10258884181386150685882078403, 6.36350501819475875124034433767, 6.81739302926739690352145119055, 7.18662181021587355110342057432, 7.25387520366286420394293603740, 7.39130873804695683583169536346, 7.73248284358295560912386242723, 7.986876294054291129943389105104