| L(s) = 1 | + 3·5-s − 4·7-s − 3·11-s + 4·13-s + 3·17-s + 19-s + 3·23-s + 5·25-s + 12·29-s + 7·31-s − 12·35-s + 37-s − 12·41-s − 8·43-s − 9·47-s + 9·49-s + 3·53-s − 9·55-s + 9·59-s + 61-s + 12·65-s + 7·67-s + 73-s + 12·77-s + 13·79-s − 24·83-s + 9·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.51·7-s − 0.904·11-s + 1.10·13-s + 0.727·17-s + 0.229·19-s + 0.625·23-s + 25-s + 2.22·29-s + 1.25·31-s − 2.02·35-s + 0.164·37-s − 1.87·41-s − 1.21·43-s − 1.31·47-s + 9/7·49-s + 0.412·53-s − 1.21·55-s + 1.17·59-s + 0.128·61-s + 1.48·65-s + 0.855·67-s + 0.117·73-s + 1.36·77-s + 1.46·79-s − 2.63·83-s + 0.976·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.584800867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.584800867\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−12.18603632272705957033828963018, −12.05721099174022547234879819740, −11.24913716824266258475995230084, −10.70185343984639502147945569607, −10.03010945048315609642531373718, −10.00067533311313666691979033844, −9.749015658413658137623813785412, −8.896270539611800093161382839532, −8.337764788676738567030829354261, −8.211079863766860731141599490917, −7.02057016779190290075933797540, −6.69695841326881302583476853477, −6.30770230010870993556249534334, −5.74162173220900054190464728814, −5.19497382469695529042734941497, −4.60851187715082967625889052792, −3.35586292307381058452228809307, −3.16409515102708516884592284828, −2.32080880964095605130769555739, −1.09900860833751981997528731422,
1.09900860833751981997528731422, 2.32080880964095605130769555739, 3.16409515102708516884592284828, 3.35586292307381058452228809307, 4.60851187715082967625889052792, 5.19497382469695529042734941497, 5.74162173220900054190464728814, 6.30770230010870993556249534334, 6.69695841326881302583476853477, 7.02057016779190290075933797540, 8.211079863766860731141599490917, 8.337764788676738567030829354261, 8.896270539611800093161382839532, 9.749015658413658137623813785412, 10.00067533311313666691979033844, 10.03010945048315609642531373718, 10.70185343984639502147945569607, 11.24913716824266258475995230084, 12.05721099174022547234879819740, 12.18603632272705957033828963018
