| L(s) = 1 | − 2·4-s − 4·9-s + 3·16-s + 2·29-s + 8·36-s − 2·41-s − 4·49-s − 2·61-s − 4·64-s + 10·81-s + 2·89-s − 2·101-s + 2·109-s − 4·116-s + 4·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 4·164-s + 167-s + 169-s + ⋯ |
| L(s) = 1 | − 2·4-s − 4·9-s + 3·16-s + 2·29-s + 8·36-s − 2·41-s − 4·49-s − 2·61-s − 4·64-s + 10·81-s + 2·89-s − 2·101-s + 2·109-s − 4·116-s + 4·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 4·164-s + 167-s + 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2910836402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2910836402\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 41 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 97 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
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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
−6.41578702818397139456434879587, −6.14759707981169934111053193586, −6.01201859067437568955349665841, −5.94066651665368945124182990021, −5.77882255759706182447512753027, −5.38646890034592018374366773567, −5.28402419773630207685451165134, −4.97651031618491700260342894964, −4.86412263691690495300321104041, −4.80072268899660433167958533506, −4.65568841957767370800265542642, −4.18288672127242375762140413149, −4.10149693836474116741102207030, −3.46840180422705486251243593380, −3.44772444460552869119794460845, −3.27777613987017360413116688629, −3.13289193927527128182397557589, −3.04799198685759782132689532728, −2.60856274578672415848975273332, −2.43083820172957114867873953669, −1.94002225091826912362541592745, −1.67676929632192092176702907719, −1.32037444264783487904565903411, −0.62828097749147127048723355992, −0.40955603311560012748092629782,
0.40955603311560012748092629782, 0.62828097749147127048723355992, 1.32037444264783487904565903411, 1.67676929632192092176702907719, 1.94002225091826912362541592745, 2.43083820172957114867873953669, 2.60856274578672415848975273332, 3.04799198685759782132689532728, 3.13289193927527128182397557589, 3.27777613987017360413116688629, 3.44772444460552869119794460845, 3.46840180422705486251243593380, 4.10149693836474116741102207030, 4.18288672127242375762140413149, 4.65568841957767370800265542642, 4.80072268899660433167958533506, 4.86412263691690495300321104041, 4.97651031618491700260342894964, 5.28402419773630207685451165134, 5.38646890034592018374366773567, 5.77882255759706182447512753027, 5.94066651665368945124182990021, 6.01201859067437568955349665841, 6.14759707981169934111053193586, 6.41578702818397139456434879587
