L(s) = 1 | + 4-s + 9-s + 4·11-s − 3·16-s + 8·23-s + 36-s + 4·37-s + 4·44-s + 8·47-s + 2·49-s − 12·53-s − 8·59-s − 7·64-s + 16·67-s + 81-s + 20·89-s + 8·92-s − 4·97-s + 4·99-s + 8·103-s − 20·113-s + 5·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 1/3·9-s + 1.20·11-s − 3/4·16-s + 1.66·23-s + 1/6·36-s + 0.657·37-s + 0.603·44-s + 1.16·47-s + 2/7·49-s − 1.64·53-s − 1.04·59-s − 7/8·64-s + 1.95·67-s + 1/9·81-s + 2.11·89-s + 0.834·92-s − 0.406·97-s + 0.402·99-s + 0.788·103-s − 1.88·113-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.610046949\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.610046949\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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
−8.321901410983147784782225497390, −7.79160688317735703649386106784, −7.41503978062230411228249840304, −6.90228927774981096827430868561, −6.54691867128892398380448569198, −6.31836306406027269916449656910, −5.62209184456279936005959921314, −5.05973013311856757256619157023, −4.57353477112992819775558845460, −4.12133772935855038399314222127, −3.50557127983327830356815345787, −2.94483568739296911207035182043, −2.29986169918340843987310771597, −1.59689334500296750702424700098, −0.851474115234783540267354877413,
0.851474115234783540267354877413, 1.59689334500296750702424700098, 2.29986169918340843987310771597, 2.94483568739296911207035182043, 3.50557127983327830356815345787, 4.12133772935855038399314222127, 4.57353477112992819775558845460, 5.05973013311856757256619157023, 5.62209184456279936005959921314, 6.31836306406027269916449656910, 6.54691867128892398380448569198, 6.90228927774981096827430868561, 7.41503978062230411228249840304, 7.79160688317735703649386106784, 8.321901410983147784782225497390