| L(s) = 1 | + 4·13-s − 4·31-s − 4·43-s + 2·49-s + 4·67-s + 4·97-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 4·13-s − 4·31-s − 4·43-s + 2·49-s + 4·67-s + 4·97-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.025228224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.025228224\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.11411862006949073534920299243, −5.99032145995315061026829096978, −5.91461776435213712864891445784, −5.85472688345247317678854119381, −5.48714092260932165488315343289, −5.23697828147467300500302241865, −5.17149725837332307746917214641, −4.85049348859462938965811847660, −4.82831800522307936409262953472, −4.30517895314614980040945814815, −4.20549152254513557178226552028, −3.83758255451358024999713724669, −3.69074829494370342818654439173, −3.57164705183215532076690067792, −3.34963672849255140260452175914, −3.34579123097174486851651422760, −3.29158392515853272049768848592, −2.42081441158556368020199952387, −2.35703274705881223463641250140, −2.15059653316745796739239460530, −1.73380066805752416871519918833, −1.63504707328161494669329486038, −1.31873431063958502237368471649, −1.01253875435238565152241279947, −0.58022039473664293180771369122,
0.58022039473664293180771369122, 1.01253875435238565152241279947, 1.31873431063958502237368471649, 1.63504707328161494669329486038, 1.73380066805752416871519918833, 2.15059653316745796739239460530, 2.35703274705881223463641250140, 2.42081441158556368020199952387, 3.29158392515853272049768848592, 3.34579123097174486851651422760, 3.34963672849255140260452175914, 3.57164705183215532076690067792, 3.69074829494370342818654439173, 3.83758255451358024999713724669, 4.20549152254513557178226552028, 4.30517895314614980040945814815, 4.82831800522307936409262953472, 4.85049348859462938965811847660, 5.17149725837332307746917214641, 5.23697828147467300500302241865, 5.48714092260932165488315343289, 5.85472688345247317678854119381, 5.91461776435213712864891445784, 5.99032145995315061026829096978, 6.11411862006949073534920299243
