| L(s) = 1 | − 4·2-s + 4·4-s + 16·8-s − 14·11-s − 64·16-s + 4·17-s + 68·19-s + 56·22-s − 50·25-s + 64·32-s − 16·34-s − 272·38-s + 46·41-s + 14·43-s − 56·44-s − 98·49-s + 200·50-s + 82·59-s + 192·64-s + 62·67-s + 16·68-s + 284·73-s + 272·76-s − 184·82-s + 316·83-s − 56·86-s − 224·88-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 2·8-s − 1.27·11-s − 4·16-s + 4/17·17-s + 3.57·19-s + 2.54·22-s − 2·25-s + 2·32-s − 0.470·34-s − 7.15·38-s + 1.12·41-s + 0.325·43-s − 1.27·44-s − 2·49-s + 4·50-s + 1.38·59-s + 3·64-s + 0.925·67-s + 4/17·68-s + 3.89·73-s + 3.57·76-s − 2.24·82-s + 3.80·83-s − 0.651·86-s − 2.54·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3879453166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3879453166\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2}( 1 - 14 T + 75 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T - 285 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T + 795 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2}( 1 + 46 T + 435 T^{2} + 46 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2}( 1 + 14 T - 1653 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2}( 1 + 82 T + 3243 T^{2} + 82 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2}( 1 + 62 T - 645 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 142 T + 14835 T^{2} - 142 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 158 T + 18075 T^{2} - 158 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 146 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{2}( 1 - 94 T - 573 T^{2} - 94 p^{2} T^{3} + p^{4} T^{4} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.609023505265173200641283558035, −8.539279138948929880449458575976, −8.043960930087855465230227610986, −8.030410553084850833720197251104, −8.024841636888128569183503125444, −7.45278819079553199638414103337, −7.33145222187899557169835312468, −7.29925071790697913586880597815, −6.73608954010980350475924301056, −6.52412037713734530892687853162, −5.99245713687831300836265771623, −5.56110220437488905844570875122, −5.32413880187821766657740871065, −5.19427117847164033546789147460, −5.02894901456723666273411390729, −4.39196292947973552380071960533, −4.14578394285584096654554668804, −3.59972957805768996764003003245, −3.50128333322726145194408794742, −2.88931258865112784245262770458, −2.35013170716717550082801988236, −2.05168015986657823888316682441, −1.19224508629314477858381693095, −1.11253783113985404478538028219, −0.31818183640242626108762187906,
0.31818183640242626108762187906, 1.11253783113985404478538028219, 1.19224508629314477858381693095, 2.05168015986657823888316682441, 2.35013170716717550082801988236, 2.88931258865112784245262770458, 3.50128333322726145194408794742, 3.59972957805768996764003003245, 4.14578394285584096654554668804, 4.39196292947973552380071960533, 5.02894901456723666273411390729, 5.19427117847164033546789147460, 5.32413880187821766657740871065, 5.56110220437488905844570875122, 5.99245713687831300836265771623, 6.52412037713734530892687853162, 6.73608954010980350475924301056, 7.29925071790697913586880597815, 7.33145222187899557169835312468, 7.45278819079553199638414103337, 8.024841636888128569183503125444, 8.030410553084850833720197251104, 8.043960930087855465230227610986, 8.539279138948929880449458575976, 8.609023505265173200641283558035
