| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s − 3·9-s + 6·12-s + 4·13-s + 5·16-s + 6·17-s + 6·18-s − 8·19-s + 8·23-s − 8·24-s − 5·25-s − 8·26-s − 14·27-s + 8·29-s + 8·31-s − 6·32-s − 12·34-s − 9·36-s − 8·37-s + 16·38-s + 8·39-s − 2·41-s + 18·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s − 9-s + 1.73·12-s + 1.10·13-s + 5/4·16-s + 1.45·17-s + 1.41·18-s − 1.83·19-s + 1.66·23-s − 1.63·24-s − 25-s − 1.56·26-s − 2.69·27-s + 1.48·29-s + 1.43·31-s − 1.06·32-s − 2.05·34-s − 3/2·36-s − 1.31·37-s + 2.59·38-s + 1.28·39-s − 0.312·41-s + 2.74·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.405426488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.405426488\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 69 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 73 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 141 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 137 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−8.534285512219681686640839773583, −8.355764237564569755882687317668, −8.019288952978361809188703318114, −8.014531635290879430218830283012, −7.14368330637273861411458337256, −7.09188630156887737401962544895, −6.35007288105815864407310006004, −6.33649260424057826301408369667, −5.79095007655999394355797933529, −5.43522957342959317052019478745, −4.99353030310726807674804620542, −4.32951795469305050456659829024, −3.68298249363124052682130841652, −3.49748352197124856107993980897, −3.08744596869431628828716120223, −2.48939869980901453179050422761, −2.31097183071017555752651366330, −1.79548021018250927903101003705, −0.77101674208666584915683031833, −0.76825699300809943226227907023,
0.76825699300809943226227907023, 0.77101674208666584915683031833, 1.79548021018250927903101003705, 2.31097183071017555752651366330, 2.48939869980901453179050422761, 3.08744596869431628828716120223, 3.49748352197124856107993980897, 3.68298249363124052682130841652, 4.32951795469305050456659829024, 4.99353030310726807674804620542, 5.43522957342959317052019478745, 5.79095007655999394355797933529, 6.33649260424057826301408369667, 6.35007288105815864407310006004, 7.09188630156887737401962544895, 7.14368330637273861411458337256, 8.014531635290879430218830283012, 8.019288952978361809188703318114, 8.355764237564569755882687317668, 8.534285512219681686640839773583
