| L(s) = 1 | − 4-s − 10·7-s + 20·13-s − 15·16-s − 32·19-s + 10·28-s − 2·31-s − 40·37-s − 100·43-s − 23·49-s − 20·52-s − 152·61-s + 31·64-s + 20·67-s − 130·73-s + 32·76-s + 28·79-s − 200·91-s + 170·97-s − 340·103-s + 328·109-s + 150·112-s + 17·121-s + 2·124-s + 127-s + 131-s + 320·133-s + ⋯ |
| L(s) = 1 | − 1/4·4-s − 1.42·7-s + 1.53·13-s − 0.937·16-s − 1.68·19-s + 5/14·28-s − 0.0645·31-s − 1.08·37-s − 2.32·43-s − 0.469·49-s − 0.384·52-s − 2.49·61-s + 0.484·64-s + 0.298·67-s − 1.78·73-s + 8/19·76-s + 0.354·79-s − 2.19·91-s + 1.75·97-s − 3.30·103-s + 3.00·109-s + 1.33·112-s + 0.140·121-s + 1/62·124-s + 0.00787·127-s + 0.00763·131-s + 2.40·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3256322348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3256322348\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 254 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 914 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 782 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 238 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4382 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4889 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6062 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 76 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 1982 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 65 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13769 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 7742 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 85 T + p^{2} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.67587915221533355837769401763, −9.905383218771869106528944749484, −9.796962122557018960023864436899, −9.078106727794841488382390915380, −8.892417280692896268033473812637, −8.310608534702233265147998582530, −8.238575808576950760502637027790, −7.26154651414221082166863844826, −6.87053137633457994467497165525, −6.52196628019785960255477010365, −5.98483552622842742705318513671, −5.91790289059649036004461818895, −4.74169295307376929737084033516, −4.69737005364622446445687101251, −3.83131045327827886133827646453, −3.42607753312031372266564999915, −3.01681557078128616562230795573, −2.07781734025838656077559329272, −1.48195358590091850510617635289, −0.20312955837779078798038045832,
0.20312955837779078798038045832, 1.48195358590091850510617635289, 2.07781734025838656077559329272, 3.01681557078128616562230795573, 3.42607753312031372266564999915, 3.83131045327827886133827646453, 4.69737005364622446445687101251, 4.74169295307376929737084033516, 5.91790289059649036004461818895, 5.98483552622842742705318513671, 6.52196628019785960255477010365, 6.87053137633457994467497165525, 7.26154651414221082166863844826, 8.238575808576950760502637027790, 8.310608534702233265147998582530, 8.892417280692896268033473812637, 9.078106727794841488382390915380, 9.796962122557018960023864436899, 9.905383218771869106528944749484, 10.67587915221533355837769401763
