| L(s) = 1 | − 3-s − 3·4-s + 8·7-s + 9-s + 3·12-s + 4·13-s + 5·16-s − 2·19-s − 8·21-s + 25-s − 27-s − 24·28-s − 3·36-s − 12·37-s − 4·39-s + 16·43-s − 5·48-s + 34·49-s − 12·52-s + 2·57-s + 28·61-s + 8·63-s − 3·64-s − 8·67-s − 28·73-s − 75-s + 6·76-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s + 3.02·7-s + 1/3·9-s + 0.866·12-s + 1.10·13-s + 5/4·16-s − 0.458·19-s − 1.74·21-s + 1/5·25-s − 0.192·27-s − 4.53·28-s − 1/2·36-s − 1.97·37-s − 0.640·39-s + 2.43·43-s − 0.721·48-s + 34/7·49-s − 1.66·52-s + 0.264·57-s + 3.58·61-s + 1.00·63-s − 3/8·64-s − 0.977·67-s − 3.27·73-s − 0.115·75-s + 0.688·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.533147475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533147475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.956131697052494305782515642843, −8.576861436588155713465515075006, −8.056447043933612031171269197880, −7.79111029211693930669614606595, −7.22090991652715915706448156971, −6.54655367252788500605108631348, −5.59885321084019164088416378698, −5.57442263951836682011583202205, −4.80915338219819536204131275847, −4.73512523477553586192350141458, −4.00775128266596845974693279239, −3.75150945152824378664368493386, −2.32830304228468473527730093464, −1.55903104009719190466896724014, −0.908717833826787650120905118138,
0.908717833826787650120905118138, 1.55903104009719190466896724014, 2.32830304228468473527730093464, 3.75150945152824378664368493386, 4.00775128266596845974693279239, 4.73512523477553586192350141458, 4.80915338219819536204131275847, 5.57442263951836682011583202205, 5.59885321084019164088416378698, 6.54655367252788500605108631348, 7.22090991652715915706448156971, 7.79111029211693930669614606595, 8.056447043933612031171269197880, 8.576861436588155713465515075006, 8.956131697052494305782515642843
