| L(s) = 1 | − 6·3-s + 16·9-s + 4·13-s − 6·17-s − 6·19-s + 6·23-s − 14·25-s − 24·27-s + 12·29-s − 8·37-s − 24·39-s − 18·41-s − 36·43-s − 10·49-s + 36·51-s + 36·57-s − 12·59-s + 36·61-s − 6·67-s − 36·69-s − 30·71-s + 84·75-s + 24·79-s + 21·81-s − 24·83-s − 72·87-s − 12·89-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 16/3·9-s + 1.10·13-s − 1.45·17-s − 1.37·19-s + 1.25·23-s − 2.79·25-s − 4.61·27-s + 2.22·29-s − 1.31·37-s − 3.84·39-s − 2.81·41-s − 5.48·43-s − 1.42·49-s + 5.04·51-s + 4.76·57-s − 1.56·59-s + 4.60·61-s − 0.733·67-s − 4.33·69-s − 3.56·71-s + 9.69·75-s + 2.70·79-s + 7/3·81-s − 2.63·83-s − 7.71·87-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1405098498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1405098498\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 p T + 20 T^{2} + 16 p T^{3} + 91 T^{4} + 16 p^{2} T^{5} + 20 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 18 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T - 8 T^{2} + 36 T^{3} + 891 T^{4} + 36 p T^{5} - 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - 16 T^{2} - 36 T^{3} + 1347 T^{4} - 36 p T^{5} - 16 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 12 T + 109 T^{2} - 732 T^{3} + 4272 T^{4} - 732 p T^{5} + 109 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 678 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + T^{2} - 88 T^{3} + 232 T^{4} - 88 p T^{5} + p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 36 T + 622 T^{2} + 6840 T^{3} + 52827 T^{4} + 6840 p T^{5} + 622 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 1818 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 1899 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 38 T^{2} - 144 T^{3} - 741 T^{4} - 144 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 36 T + 653 T^{2} - 7956 T^{3} + 71472 T^{4} - 7956 p T^{5} + 653 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T - 80 T^{2} - 108 T^{3} + 6555 T^{4} - 108 p T^{5} - 80 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 30 T + 508 T^{2} + 6240 T^{3} + 59523 T^{4} + 6240 p T^{5} + 508 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 94 T^{2} + 552 T^{3} - 1533 T^{4} + 552 p T^{5} + 94 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.03366007459414717175505264081, −6.89996849363826723968302867436, −6.83888435743621444972540711504, −6.57360282905079723222950624988, −6.40636232791891529767911468965, −6.17560028935469063834556694163, −6.03931729273489295845264799227, −5.53209272287889459887740227687, −5.47007774894861552043292521614, −5.33961648966361732952469451127, −5.26524818147576124065907596841, −4.72668638025934901512950445731, −4.58934915258781921975665968458, −4.34557721680566296593583925808, −4.32168144835051336123744765097, −3.67942922941807242589105191151, −3.46179129825555789843229820554, −3.10090611796105007247769626061, −3.04721831298631488284570755128, −2.24988707983899543841508529829, −1.82910717526818829491881239249, −1.55733465656631249562244936218, −1.51583098123029121049522291003, −0.38385948736561622085202010658, −0.29863194188471402552217509324,
0.29863194188471402552217509324, 0.38385948736561622085202010658, 1.51583098123029121049522291003, 1.55733465656631249562244936218, 1.82910717526818829491881239249, 2.24988707983899543841508529829, 3.04721831298631488284570755128, 3.10090611796105007247769626061, 3.46179129825555789843229820554, 3.67942922941807242589105191151, 4.32168144835051336123744765097, 4.34557721680566296593583925808, 4.58934915258781921975665968458, 4.72668638025934901512950445731, 5.26524818147576124065907596841, 5.33961648966361732952469451127, 5.47007774894861552043292521614, 5.53209272287889459887740227687, 6.03931729273489295845264799227, 6.17560028935469063834556694163, 6.40636232791891529767911468965, 6.57360282905079723222950624988, 6.83888435743621444972540711504, 6.89996849363826723968302867436, 7.03366007459414717175505264081
