| L(s) = 1 | + 4·49-s − 81-s − 8·97-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | + 4·49-s − 81-s − 8·97-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.373027916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.373027916\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$ | \( ( 1 + T )^{8} \) |
| 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
−6.42785414255528850372055491136, −5.94094145167146399279470691750, −5.84708554138472942478538196571, −5.75452046609404865511276035549, −5.70008998277433213533701522031, −5.43063973870499996027286541794, −5.12709580101326776447169180230, −4.98295731119253467662722568788, −4.81478377608992146015288375312, −4.32620683929509160746485781991, −4.24377471230616338001811570280, −4.16230907033899008918432432400, −3.89702041163973258740634789180, −3.77662218771910359513202118686, −3.48088156479517302531132840367, −3.06496364151215791319726696955, −2.74198755636911149032372621669, −2.72425410088202730598982268643, −2.68939251459845200889645947595, −2.15055032143309626007799534134, −1.95712338329204378325968463816, −1.53350743937574894770540309698, −1.36725538656361626900124952109, −0.994218263864976680754827617339, −0.51063105119599922290331032342,
0.51063105119599922290331032342, 0.994218263864976680754827617339, 1.36725538656361626900124952109, 1.53350743937574894770540309698, 1.95712338329204378325968463816, 2.15055032143309626007799534134, 2.68939251459845200889645947595, 2.72425410088202730598982268643, 2.74198755636911149032372621669, 3.06496364151215791319726696955, 3.48088156479517302531132840367, 3.77662218771910359513202118686, 3.89702041163973258740634789180, 4.16230907033899008918432432400, 4.24377471230616338001811570280, 4.32620683929509160746485781991, 4.81478377608992146015288375312, 4.98295731119253467662722568788, 5.12709580101326776447169180230, 5.43063973870499996027286541794, 5.70008998277433213533701522031, 5.75452046609404865511276035549, 5.84708554138472942478538196571, 5.94094145167146399279470691750, 6.42785414255528850372055491136
