L(s) = 1 | − 3-s + 3·5-s + 7-s + 3·11-s − 3·15-s + 6·17-s + 2·19-s − 21-s + 25-s + 27-s + 18·29-s − 5·31-s − 3·33-s + 3·35-s − 10·37-s − 12·47-s − 6·49-s − 6·51-s + 9·53-s + 9·55-s − 2·57-s + 9·59-s − 24·67-s − 12·73-s − 75-s + 3·77-s + 9·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.377·7-s + 0.904·11-s − 0.774·15-s + 1.45·17-s + 0.458·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 3.34·29-s − 0.898·31-s − 0.522·33-s + 0.507·35-s − 1.64·37-s − 1.75·47-s − 6/7·49-s − 0.840·51-s + 1.23·53-s + 1.21·55-s − 0.264·57-s + 1.17·59-s − 2.93·67-s − 1.40·73-s − 0.115·75-s + 0.341·77-s + 1.01·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.917857798\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.917857798\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 24 T + 259 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T + 101 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−11.86955287888147306674300484446, −11.44745381110301718548587896288, −10.63515159240329913830380747356, −10.46377590517617138320775756959, −9.819315281793336142955598128581, −9.776979601332275083691057184293, −9.031441912788082727672425083054, −8.531191304456363468334837613519, −8.138714065745209770547273243674, −7.42819210345826005041097765573, −6.65753218298887524195105826938, −6.62133921340766585861987217264, −5.68545939867883456956698803485, −5.63013305515793625995455845003, −4.91742076453252468906770468423, −4.38546512750434789043718867075, −3.41515829207316476287489313638, −2.86885285692459854617263203460, −1.76568093759239594602061038249, −1.19226971877493348484139312883,
1.19226971877493348484139312883, 1.76568093759239594602061038249, 2.86885285692459854617263203460, 3.41515829207316476287489313638, 4.38546512750434789043718867075, 4.91742076453252468906770468423, 5.63013305515793625995455845003, 5.68545939867883456956698803485, 6.62133921340766585861987217264, 6.65753218298887524195105826938, 7.42819210345826005041097765573, 8.138714065745209770547273243674, 8.531191304456363468334837613519, 9.031441912788082727672425083054, 9.776979601332275083691057184293, 9.819315281793336142955598128581, 10.46377590517617138320775756959, 10.63515159240329913830380747356, 11.44745381110301718548587896288, 11.86955287888147306674300484446