| L(s) = 1 | − 4·5-s − 6·9-s − 4·13-s − 12·17-s + 2·25-s − 12·29-s + 4·37-s + 4·41-s + 24·45-s + 49-s − 12·53-s + 12·61-s + 16·65-s + 20·73-s + 27·81-s + 48·85-s − 12·89-s − 12·97-s − 4·101-s + 20·109-s + 4·113-s + 24·117-s − 6·121-s + 28·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 2·9-s − 1.10·13-s − 2.91·17-s + 2/5·25-s − 2.22·29-s + 0.657·37-s + 0.624·41-s + 3.57·45-s + 1/7·49-s − 1.64·53-s + 1.53·61-s + 1.98·65-s + 2.34·73-s + 3·81-s + 5.20·85-s − 1.27·89-s − 1.21·97-s − 0.398·101-s + 1.91·109-s + 0.376·113-s + 2.21·117-s − 0.545·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.749304120934004278681759852831, −8.056242270008326333596348707107, −7.79292083571821938290976650763, −7.26382021666693896035142333427, −6.78746217059840482908609994194, −6.15054397459112152814277916997, −5.69394971603405508863657314834, −4.92225264716551837532790929786, −4.58367495132727840614913978641, −3.77515777589665000060106383714, −3.60173413237973468362473708205, −2.40145156924384381758797773878, −2.35911554344998475560129433401, 0, 0,
2.35911554344998475560129433401, 2.40145156924384381758797773878, 3.60173413237973468362473708205, 3.77515777589665000060106383714, 4.58367495132727840614913978641, 4.92225264716551837532790929786, 5.69394971603405508863657314834, 6.15054397459112152814277916997, 6.78746217059840482908609994194, 7.26382021666693896035142333427, 7.79292083571821938290976650763, 8.056242270008326333596348707107, 8.749304120934004278681759852831
