| L(s) = 1 | − 4·3-s + 4·7-s + 6·9-s + 16·13-s − 16·21-s + 2·25-s + 4·27-s − 64·39-s − 2·49-s + 24·63-s − 32·73-s − 8·75-s − 32·79-s − 37·81-s + 64·91-s − 32·97-s − 40·103-s − 40·109-s + 96·117-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 8·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.51·7-s + 2·9-s + 4.43·13-s − 3.49·21-s + 2/5·25-s + 0.769·27-s − 10.2·39-s − 2/7·49-s + 3.02·63-s − 3.74·73-s − 0.923·75-s − 3.60·79-s − 4.11·81-s + 6.70·91-s − 3.24·97-s − 3.94·103-s − 3.83·109-s + 8.87·117-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.659·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.501241946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.501241946\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 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
−6.63844157156451038617884054003, −6.35144895167287782233063438713, −6.16508960360217832343007898580, −5.86719616877824092674159644256, −5.82056735183562409944732391328, −5.66917371693870402910452968101, −5.57554421801130637442910554631, −5.17124065935850563394472220571, −5.02400551700898637918425369742, −4.80859884553317516933557605745, −4.54094572583977094282012943181, −4.18604653068758034872309553944, −4.03232827394358621798409958550, −3.93835278529017192382981231900, −3.79201205800946477740482840659, −3.22412714298363682565510952801, −2.84270818125192484025098966162, −2.77954387174995072395431218013, −2.73402507758405591985094099117, −1.57356564596574809741445957369, −1.53594362494527762656225120554, −1.46572922768665464353829004887, −1.39875847841893684697161315665, −0.78101735804922216981723828745, −0.31977115304105134625154788386,
0.31977115304105134625154788386, 0.78101735804922216981723828745, 1.39875847841893684697161315665, 1.46572922768665464353829004887, 1.53594362494527762656225120554, 1.57356564596574809741445957369, 2.73402507758405591985094099117, 2.77954387174995072395431218013, 2.84270818125192484025098966162, 3.22412714298363682565510952801, 3.79201205800946477740482840659, 3.93835278529017192382981231900, 4.03232827394358621798409958550, 4.18604653068758034872309553944, 4.54094572583977094282012943181, 4.80859884553317516933557605745, 5.02400551700898637918425369742, 5.17124065935850563394472220571, 5.57554421801130637442910554631, 5.66917371693870402910452968101, 5.82056735183562409944732391328, 5.86719616877824092674159644256, 6.16508960360217832343007898580, 6.35144895167287782233063438713, 6.63844157156451038617884054003
