L(s) = 1 | − 3·3-s − 2·4-s − 5-s − 5·7-s + 6·9-s + 3·11-s + 6·12-s + 3·15-s − 12·17-s + 2·20-s + 15·21-s + 6·23-s − 9·27-s + 10·28-s − 15·29-s + 6·31-s − 9·33-s + 5·35-s − 12·36-s − 16·37-s − 6·41-s − 10·43-s − 6·44-s − 6·45-s − 9·47-s + 18·49-s + 36·51-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 4-s − 0.447·5-s − 1.88·7-s + 2·9-s + 0.904·11-s + 1.73·12-s + 0.774·15-s − 2.91·17-s + 0.447·20-s + 3.27·21-s + 1.25·23-s − 1.73·27-s + 1.88·28-s − 2.78·29-s + 1.07·31-s − 1.56·33-s + 0.845·35-s − 2·36-s − 2.63·37-s − 0.937·41-s − 1.52·43-s − 0.904·44-s − 0.894·45-s − 1.31·47-s + 18/7·49-s + 5.04·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T + 104 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 139 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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
−11.32676804656343459989476409285, −11.11354614702334595031068555394, −10.56051525568227247938542916333, −10.11207876206754506228091195866, −9.359240673800704858865599609601, −9.331098130362485228446961979221, −8.820354168012346425549062737348, −8.278196322979109155403078955892, −7.05289880414395618496727379726, −6.81301939619650145705207178865, −6.72535267588907873777839382021, −6.13531485755614738198360963806, −5.16626329699123585410935558924, −5.13231194078843606176249244297, −4.06201787591128922717530840258, −4.04357415347202830850886672646, −3.15037463265325187270500173693, −1.78197123018775952764122594914, 0, 0,
1.78197123018775952764122594914, 3.15037463265325187270500173693, 4.04357415347202830850886672646, 4.06201787591128922717530840258, 5.13231194078843606176249244297, 5.16626329699123585410935558924, 6.13531485755614738198360963806, 6.72535267588907873777839382021, 6.81301939619650145705207178865, 7.05289880414395618496727379726, 8.278196322979109155403078955892, 8.820354168012346425549062737348, 9.331098130362485228446961979221, 9.359240673800704858865599609601, 10.11207876206754506228091195866, 10.56051525568227247938542916333, 11.11354614702334595031068555394, 11.32676804656343459989476409285