| L(s) = 1 | + 4-s + 2·11-s − 2·29-s + 2·44-s − 49-s − 64-s + 2·71-s + 2·79-s + 2·109-s − 2·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 196-s + ⋯ |
| L(s) = 1 | + 4-s + 2·11-s − 2·29-s + 2·44-s − 49-s − 64-s + 2·71-s + 2·79-s + 2·109-s − 2·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.623019609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.623019609\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.697831863137748353464176588330, −9.477065285462139530277909108834, −8.976846395961361110782148213354, −8.808036408677867291682335091912, −8.205357190835298406048859052435, −7.66476373088840608831281575575, −7.42557907845317365135588293843, −6.96840610634510930146297951669, −6.52929929537319611138161596603, −6.27966694123544293748218645752, −6.01379839869926721671148843739, −5.21546728565667259673098134671, −5.00286332828853934429028089546, −4.20615326396593357826942978377, −3.82671895408862407941536656031, −3.51480041582539475256755452809, −2.87465698894886530916011711200, −2.08086484378021364477569432625, −1.84764916556910467273797050135, −1.09915149154151201136701894336,
1.09915149154151201136701894336, 1.84764916556910467273797050135, 2.08086484378021364477569432625, 2.87465698894886530916011711200, 3.51480041582539475256755452809, 3.82671895408862407941536656031, 4.20615326396593357826942978377, 5.00286332828853934429028089546, 5.21546728565667259673098134671, 6.01379839869926721671148843739, 6.27966694123544293748218645752, 6.52929929537319611138161596603, 6.96840610634510930146297951669, 7.42557907845317365135588293843, 7.66476373088840608831281575575, 8.205357190835298406048859052435, 8.808036408677867291682335091912, 8.976846395961361110782148213354, 9.477065285462139530277909108834, 9.697831863137748353464176588330
