| L(s) = 1 | + 5-s − 9-s − 2·13-s − 3·37-s + 2·41-s − 45-s − 4·49-s + 3·53-s − 2·65-s − 3·89-s + 5·113-s + 2·117-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s − 3·185-s + ⋯ |
| L(s) = 1 | + 5-s − 9-s − 2·13-s − 3·37-s + 2·41-s − 45-s − 4·49-s + 3·53-s − 2·65-s − 3·89-s + 5·113-s + 2·117-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s − 3·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9750041562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9750041562\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| good | 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 13 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 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
−6.21874137446378448677823162967, −6.12022973901593065669869083454, −6.04255598830552611057907101856, −5.53586724180165001563099815103, −5.47426917507994459284615259949, −5.47141016339847157820555801417, −5.16886497011220838195987559863, −4.94692469092921468492526216909, −4.84080151645761567502847403087, −4.46600258136884012898375722130, −4.33501812463327090338104655637, −4.22575677171895238233258594378, −3.75915319077020106994138177379, −3.48899782055941207676087143385, −3.44592323010345838663995374553, −3.08565149026226327276546611746, −2.81199641290466401922104410182, −2.74967141423410022935365826485, −2.32541086982479454478523979869, −2.30000879745002115212267968323, −1.76960635658735112133165871690, −1.75257620037380458756357523666, −1.55793908799703878535171724715, −0.855779514923479521528128194326, −0.41606393121380592324780626731,
0.41606393121380592324780626731, 0.855779514923479521528128194326, 1.55793908799703878535171724715, 1.75257620037380458756357523666, 1.76960635658735112133165871690, 2.30000879745002115212267968323, 2.32541086982479454478523979869, 2.74967141423410022935365826485, 2.81199641290466401922104410182, 3.08565149026226327276546611746, 3.44592323010345838663995374553, 3.48899782055941207676087143385, 3.75915319077020106994138177379, 4.22575677171895238233258594378, 4.33501812463327090338104655637, 4.46600258136884012898375722130, 4.84080151645761567502847403087, 4.94692469092921468492526216909, 5.16886497011220838195987559863, 5.47141016339847157820555801417, 5.47426917507994459284615259949, 5.53586724180165001563099815103, 6.04255598830552611057907101856, 6.12022973901593065669869083454, 6.21874137446378448677823162967
