| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 3·9-s − 2·11-s − 2·12-s + 16-s − 3·18-s − 2·19-s + 2·22-s + 2·24-s + 2·25-s − 4·27-s − 32-s + 4·33-s + 3·36-s + 2·38-s + 16·41-s − 8·43-s − 2·44-s − 2·48-s + 8·49-s − 2·50-s + 4·54-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 9-s − 0.603·11-s − 0.577·12-s + 1/4·16-s − 0.707·18-s − 0.458·19-s + 0.426·22-s + 0.408·24-s + 2/5·25-s − 0.769·27-s − 0.176·32-s + 0.696·33-s + 1/2·36-s + 0.324·38-s + 2.49·41-s − 1.21·43-s − 0.301·44-s − 0.288·48-s + 8/7·49-s − 0.282·50-s + 0.544·54-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6279092152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6279092152\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 120 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.914554699542904674641072552467, −8.239137892831495241953113812205, −7.79585105238335101105621439086, −7.34455704116900657750244326553, −6.99247034251610157842798307777, −6.30616171569915088687865707581, −6.05056574276622021378055246385, −5.53313292471925168840802790698, −4.97971529323042890968218479061, −4.44980477837391128126017725747, −3.88877477978908394437083093022, −3.00704719912046339113834832268, −2.37448383667542832793306086425, −1.50788827271432140661898633032, −0.56099761933216304669556331259,
0.56099761933216304669556331259, 1.50788827271432140661898633032, 2.37448383667542832793306086425, 3.00704719912046339113834832268, 3.88877477978908394437083093022, 4.44980477837391128126017725747, 4.97971529323042890968218479061, 5.53313292471925168840802790698, 6.05056574276622021378055246385, 6.30616171569915088687865707581, 6.99247034251610157842798307777, 7.34455704116900657750244326553, 7.79585105238335101105621439086, 8.239137892831495241953113812205, 8.914554699542904674641072552467
