| L(s) = 1 | + 9-s + 4·19-s + 25-s + 2·49-s + 2·53-s + 2·61-s + 4·67-s − 4·71-s − 4·73-s + 2·83-s − 2·89-s − 2·97-s − 2·103-s − 4·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 4·171-s + 173-s + ⋯ |
| L(s) = 1 | + 9-s + 4·19-s + 25-s + 2·49-s + 2·53-s + 2·61-s + 4·67-s − 4·71-s − 4·73-s + 2·83-s − 2·89-s − 2·97-s − 2·103-s − 4·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 4·171-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 487^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 487^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.941785334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.941785334\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 487 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 59 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 61 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 79 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 83 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 89 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
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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
−6.84322992757999369360685071720, −6.62113820874260692574878536017, −6.53412928717517654558846478243, −5.78437404328818324244226024929, −5.74903491201773153126487850241, −5.65842852523365832628344910831, −5.56263095487505603829012404918, −5.34414338143641816625584271192, −5.23419919197421440466735708512, −4.60070492526647162763908375564, −4.57616449926536356803576590461, −4.47652917035015507204814214535, −4.22192485645009204541912020835, −3.79755615706211329379179308465, −3.61825157888598133421614413996, −3.49274812908849543544446407010, −3.12137570052562292480300678147, −3.00211406599325705672981002167, −2.52993209888674106822147718011, −2.42450362173624351928300737920, −2.29602091332053267682543064606, −1.51014005031690499103359731046, −1.27831037570032608929680176377, −1.18997845479417645611957506406, −0.886143423589341816213514336263,
0.886143423589341816213514336263, 1.18997845479417645611957506406, 1.27831037570032608929680176377, 1.51014005031690499103359731046, 2.29602091332053267682543064606, 2.42450362173624351928300737920, 2.52993209888674106822147718011, 3.00211406599325705672981002167, 3.12137570052562292480300678147, 3.49274812908849543544446407010, 3.61825157888598133421614413996, 3.79755615706211329379179308465, 4.22192485645009204541912020835, 4.47652917035015507204814214535, 4.57616449926536356803576590461, 4.60070492526647162763908375564, 5.23419919197421440466735708512, 5.34414338143641816625584271192, 5.56263095487505603829012404918, 5.65842852523365832628344910831, 5.74903491201773153126487850241, 5.78437404328818324244226024929, 6.53412928717517654558846478243, 6.62113820874260692574878536017, 6.84322992757999369360685071720
