| L(s) = 1 | − 18·7-s − 2·13-s − 50·19-s + 78·31-s − 64·37-s − 46·43-s + 145·49-s + 146·61-s + 126·67-s − 272·73-s + 48·79-s + 36·91-s + 14·97-s + 220·103-s − 50·109-s − 96·121-s + 127-s + 131-s + 900·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 335·169-s + ⋯ |
| L(s) = 1 | − 2.57·7-s − 0.153·13-s − 2.63·19-s + 2.51·31-s − 1.72·37-s − 1.06·43-s + 2.95·49-s + 2.39·61-s + 1.88·67-s − 3.72·73-s + 0.607·79-s + 0.395·91-s + 0.144·97-s + 2.13·103-s − 0.458·109-s − 0.793·121-s + 0.00787·127-s + 0.00763·131-s + 6.76·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.98·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03911640907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03911640907\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 9 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 96 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 144 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1040 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 960 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 39 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3330 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 23 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3360 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3630 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6864 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 73 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 63 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6210 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 136 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 11600 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )( 1 + 146 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.776138061745164346003530987262, −8.252796158568762970501735270187, −7.997636132580568519118650313442, −7.17135805382902305190067607425, −6.98561126807676597054637360868, −6.65997258995402939772140314164, −6.38316475790437159515744164872, −6.09129067577034291958928698301, −5.83379030060162939759408543609, −5.01557329520137064972930054782, −4.86550632116231161790535574079, −4.19810831041191388595881267396, −3.86797625280582361817074448983, −3.49382943819530255912410999753, −3.08086048173263521894365911046, −2.44890198383562394385072468004, −2.41276336916997655515474657751, −1.57228583041082317301656131480, −0.77087749458354807650065673550, −0.05523172074852043384134555270,
0.05523172074852043384134555270, 0.77087749458354807650065673550, 1.57228583041082317301656131480, 2.41276336916997655515474657751, 2.44890198383562394385072468004, 3.08086048173263521894365911046, 3.49382943819530255912410999753, 3.86797625280582361817074448983, 4.19810831041191388595881267396, 4.86550632116231161790535574079, 5.01557329520137064972930054782, 5.83379030060162939759408543609, 6.09129067577034291958928698301, 6.38316475790437159515744164872, 6.65997258995402939772140314164, 6.98561126807676597054637360868, 7.17135805382902305190067607425, 7.997636132580568519118650313442, 8.252796158568762970501735270187, 8.776138061745164346003530987262
