| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s + 4·5-s − 4·6-s + 6·7-s + 3·9-s − 8·10-s − 8·11-s + 4·12-s + 4·13-s − 12·14-s + 8·15-s − 4·16-s − 6·18-s − 6·19-s + 8·20-s + 12·21-s + 16·22-s + 8·23-s + 8·25-s − 8·26-s + 4·27-s + 12·28-s − 16·30-s + 2·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s + 1.78·5-s − 1.63·6-s + 2.26·7-s + 9-s − 2.52·10-s − 2.41·11-s + 1.15·12-s + 1.10·13-s − 3.20·14-s + 2.06·15-s − 16-s − 1.41·18-s − 1.37·19-s + 1.78·20-s + 2.61·21-s + 3.41·22-s + 1.66·23-s + 8/5·25-s − 1.56·26-s + 0.769·27-s + 2.26·28-s − 2.92·30-s + 0.359·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.849551160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.849551160\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−11.59140910960777246297133487129, −11.03843255949456336318029551461, −10.58753583476993340298298772515, −10.55842106458327365195658126946, −10.11946079920542564162983275817, −9.484086352400828709565420899307, −8.913492157933406423399219753868, −8.507378753893691547015285373015, −8.250438907965759386514389555351, −8.121972295425459131221965947213, −7.12150507910982425926081735349, −7.07977206422364817342323710938, −6.04111746428192283709525017706, −5.31889933963681531072655976191, −4.97239467948513122547504169858, −4.48217423587517232634617200200, −3.14297076464566174802901790178, −2.40687491424041976776897269724, −1.82542213834283294130118914955, −1.49509579743978856137702165840,
1.49509579743978856137702165840, 1.82542213834283294130118914955, 2.40687491424041976776897269724, 3.14297076464566174802901790178, 4.48217423587517232634617200200, 4.97239467948513122547504169858, 5.31889933963681531072655976191, 6.04111746428192283709525017706, 7.07977206422364817342323710938, 7.12150507910982425926081735349, 8.121972295425459131221965947213, 8.250438907965759386514389555351, 8.507378753893691547015285373015, 8.913492157933406423399219753868, 9.484086352400828709565420899307, 10.11946079920542564162983275817, 10.55842106458327365195658126946, 10.58753583476993340298298772515, 11.03843255949456336318029551461, 11.59140910960777246297133487129
