| L(s) = 1 | + 6·9-s + 13·49-s + 16·79-s + 27·81-s + 68·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 46·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 2·9-s + 13/7·49-s + 1.80·79-s + 3·81-s + 6.51·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.228630694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.228630694\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 83 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 167 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.48743010089410608242476746622, −6.03837488155012154171248560544, −5.99305260347888915391722498680, −5.97629726608384868681022560228, −5.91626355514101721239787840292, −5.25711843532051393387026720537, −5.11153502246802051940146339025, −5.01410587650028801280534213848, −4.79990916377504536390143849766, −4.47763042959006819961034600339, −4.20525575450400196849538575753, −4.16644349459473959673749951746, −4.12752722639482743766759219506, −3.41222900930171949937893450307, −3.33190229275347937785287510868, −3.32085605894802316738519550674, −3.14861687764710790771742295915, −2.31236427211878930402396597464, −2.20074443196052952253487661255, −2.18285646763031811314215232010, −1.93743706417033022737543260056, −1.35482904634334075502487177937, −1.14080726101816712930625384144, −0.75950971612582345484740653022, −0.49777517793442728489038188339,
0.49777517793442728489038188339, 0.75950971612582345484740653022, 1.14080726101816712930625384144, 1.35482904634334075502487177937, 1.93743706417033022737543260056, 2.18285646763031811314215232010, 2.20074443196052952253487661255, 2.31236427211878930402396597464, 3.14861687764710790771742295915, 3.32085605894802316738519550674, 3.33190229275347937785287510868, 3.41222900930171949937893450307, 4.12752722639482743766759219506, 4.16644349459473959673749951746, 4.20525575450400196849538575753, 4.47763042959006819961034600339, 4.79990916377504536390143849766, 5.01410587650028801280534213848, 5.11153502246802051940146339025, 5.25711843532051393387026720537, 5.91626355514101721239787840292, 5.97629726608384868681022560228, 5.99305260347888915391722498680, 6.03837488155012154171248560544, 6.48743010089410608242476746622
