L(s) = 1 | + 2·3-s + 9-s − 12·11-s + 10·25-s − 4·27-s − 24·33-s + 14·49-s + 12·59-s + 4·73-s + 20·75-s − 11·81-s − 36·83-s + 20·97-s − 12·99-s + 12·107-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 28·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s − 3.61·11-s + 2·25-s − 0.769·27-s − 4.17·33-s + 2·49-s + 1.56·59-s + 0.468·73-s + 2.30·75-s − 1.22·81-s − 3.95·83-s + 2.03·97-s − 1.20·99-s + 1.16·107-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.30·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.831641843\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.831641843\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.34978052077095511992188404187, −10.08468491543284455159461073244, −9.926124375736567438467452817323, −9.014021864945821662185383300290, −8.790302036599455494425237302694, −8.301570930389600637753947718860, −8.162167699761703250175291054002, −7.49955535214414209473593430205, −7.31850640523112292206964249084, −6.89652540821914029109718029367, −5.95630682999826388559510429731, −5.52234145276171359986549740330, −5.23749201718865987480865849146, −4.69186723337989744382247992323, −4.13726076763144534312633150309, −3.19057485211952431031773564650, −2.96179654937690131647998420245, −2.49289565774882518970617759263, −2.04263476431434176938798796144, −0.60959836316426054745840675418,
0.60959836316426054745840675418, 2.04263476431434176938798796144, 2.49289565774882518970617759263, 2.96179654937690131647998420245, 3.19057485211952431031773564650, 4.13726076763144534312633150309, 4.69186723337989744382247992323, 5.23749201718865987480865849146, 5.52234145276171359986549740330, 5.95630682999826388559510429731, 6.89652540821914029109718029367, 7.31850640523112292206964249084, 7.49955535214414209473593430205, 8.162167699761703250175291054002, 8.301570930389600637753947718860, 8.790302036599455494425237302694, 9.014021864945821662185383300290, 9.926124375736567438467452817323, 10.08468491543284455159461073244, 10.34978052077095511992188404187