| L(s) = 1 | − 16·13-s + 84·25-s + 32·37-s − 148·49-s + 96·61-s − 216·73-s − 40·97-s + 352·109-s + 900·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 920·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.23·13-s + 3.35·25-s + 0.864·37-s − 3.02·49-s + 1.57·61-s − 2.95·73-s − 0.412·97-s + 3.22·109-s + 7.43·121-s + 0.00787·127-s + 0.00763·131-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 − 5.44·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.744508283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.744508283\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 42 T^{2} + 1043 T^{4} - 42 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 + 74 T^{2} + 2643 T^{4} + 74 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 450 T^{2} + 79619 T^{4} - 450 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 4 T + 270 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 - 76 T^{2} + 127014 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 980 T^{2} + 459270 T^{4} + 980 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 - 1684 T^{2} + 1227174 T^{4} - 1684 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2028 T^{2} + 2147846 T^{4} - 2028 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 646 T^{2} - 355581 T^{4} - 646 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T + 954 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 5076 T^{2} + 12019238 T^{4} - 5076 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 1916 T^{2} + 7662054 T^{4} + 1916 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 5052 T^{2} + 14107910 T^{4} - 5052 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 2538 T^{2} + 14597075 T^{4} - 2538 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 6492 T^{2} + 25440038 T^{4} - 6492 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 24 T + 2978 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 + 7676 T^{2} + 45068838 T^{4} + 7676 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 5292 T^{2} + 24922406 T^{4} - 5292 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 54 T + 2675 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 + 11660 T^{2} + 68532390 T^{4} + 11660 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 3634 T^{2} - 15003021 T^{4} - 3634 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 7332 T^{2} + 109063046 T^{4} + 7332 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 10 T + 14235 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.38742043845337020421221463021, −4.00933761571991179900017938507, −3.73555773505185841239365025369, −3.71445026583041695714130360251, −3.60249226908947890093818900983, −3.51237806713281551766023317627, −3.29708197305782856771017528863, −3.12452759142426931361158168411, −2.92662509774570393933209609099, −2.92337120602865015632877475591, −2.86565008736382194231257202476, −2.68856946886626941928521226355, −2.44391886440401997528879381507, −2.30034285779999485815555919398, −2.10052713670322295955479425390, −1.96046035274488664661615819744, −1.85788278309711083591505916152, −1.53509448152557832215379165094, −1.51324066690022645921240430737, −1.12314034361555983269032075098, −1.03482434837598114471393726289, −0.78861011338591634686672295643, −0.73480391396487983073673985239, −0.25659421368053854390674822334, −0.22276547722026821297372932774,
0.22276547722026821297372932774, 0.25659421368053854390674822334, 0.73480391396487983073673985239, 0.78861011338591634686672295643, 1.03482434837598114471393726289, 1.12314034361555983269032075098, 1.51324066690022645921240430737, 1.53509448152557832215379165094, 1.85788278309711083591505916152, 1.96046035274488664661615819744, 2.10052713670322295955479425390, 2.30034285779999485815555919398, 2.44391886440401997528879381507, 2.68856946886626941928521226355, 2.86565008736382194231257202476, 2.92337120602865015632877475591, 2.92662509774570393933209609099, 3.12452759142426931361158168411, 3.29708197305782856771017528863, 3.51237806713281551766023317627, 3.60249226908947890093818900983, 3.71445026583041695714130360251, 3.73555773505185841239365025369, 4.00933761571991179900017938507, 4.38742043845337020421221463021
Plot not available for L-functions of degree greater than 10.
