| L(s) = 1 | + 3·2-s + 3·3-s + 2·4-s + 9·6-s + 2·7-s − 3·8-s − 3·11-s + 6·12-s − 4·13-s + 6·14-s − 3·16-s + 22·19-s + 6·21-s − 9·22-s + 48·23-s − 9·24-s − 12·26-s − 27·27-s + 4·28-s + 78·29-s − 32·31-s − 12·32-s − 9·33-s + 68·37-s + 66·38-s − 12·39-s − 21·41-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 3-s + 1/2·4-s + 3/2·6-s + 2/7·7-s − 3/8·8-s − 0.272·11-s + 1/2·12-s − 0.307·13-s + 3/7·14-s − 0.187·16-s + 1.15·19-s + 2/7·21-s − 0.409·22-s + 2.08·23-s − 3/8·24-s − 0.461·26-s − 27-s + 1/7·28-s + 2.68·29-s − 1.03·31-s − 3/8·32-s − 0.272·33-s + 1.83·37-s + 1.73·38-s − 0.307·39-s − 0.512·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.727271539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.727271539\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 124 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T - 153 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 335 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 48 T + 1297 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 78 T + 2869 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 32 T + 63 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 21 T + 1828 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 83 T + p^{2} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 84 T + 4561 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 87 T + 6004 T^{2} - 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 56 T - 585 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 31 T - 3528 T^{2} + 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9110 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 65 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 38 T - 4797 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 84 T + 9241 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 115 T + 3816 T^{2} + 115 p^{2} T^{3} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.29393734284525641654382817267, −12.08563668505458177463770053434, −11.44499588051008450368502175883, −10.95724741286907687416573676431, −10.36472887676441183360150331194, −9.804021966826466590322890898455, −9.254160013058450251150141476439, −8.704173037189104765036271333712, −8.475368730805638510770018638452, −7.60851257714479647754373166190, −7.31245420871437079817718247132, −6.64838959079752674915052376570, −5.72223803823599120947494984299, −5.40575223782693855033799327934, −4.69428680709996020803998325662, −4.41454745540546679310131596477, −3.48876512101936353962441084554, −3.01689314614375536750268665932, −2.48899492490282734136567817100, −1.08431541049391403651662053484,
1.08431541049391403651662053484, 2.48899492490282734136567817100, 3.01689314614375536750268665932, 3.48876512101936353962441084554, 4.41454745540546679310131596477, 4.69428680709996020803998325662, 5.40575223782693855033799327934, 5.72223803823599120947494984299, 6.64838959079752674915052376570, 7.31245420871437079817718247132, 7.60851257714479647754373166190, 8.475368730805638510770018638452, 8.704173037189104765036271333712, 9.254160013058450251150141476439, 9.804021966826466590322890898455, 10.36472887676441183360150331194, 10.95724741286907687416573676431, 11.44499588051008450368502175883, 12.08563668505458177463770053434, 12.29393734284525641654382817267
