| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s + 8-s + 6·9-s − 3·11-s − 3·12-s + 16-s + 6·18-s − 5·19-s − 3·22-s − 3·24-s + 3·25-s − 9·27-s + 32-s + 9·33-s + 6·36-s − 5·38-s − 3·41-s + 10·43-s − 3·44-s − 3·48-s − 8·49-s + 3·50-s − 9·54-s + 15·57-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s + 0.353·8-s + 2·9-s − 0.904·11-s − 0.866·12-s + 1/4·16-s + 1.41·18-s − 1.14·19-s − 0.639·22-s − 0.612·24-s + 3/5·25-s − 1.73·27-s + 0.176·32-s + 1.56·33-s + 36-s − 0.811·38-s − 0.468·41-s + 1.52·43-s − 0.452·44-s − 0.433·48-s − 8/7·49-s + 0.424·50-s − 1.22·54-s + 1.98·57-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 51 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
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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
−8.604969221559244333634630913098, −7.69823812494879928556312846174, −7.59776998434374205435069602344, −6.89865749107197294369399543033, −6.45782929427348210743724639109, −6.13709332444057347666756704097, −5.65045951154400845042854387589, −5.10870910841514021482586673894, −4.81103346199117845966800164034, −4.28088665839059093669700037659, −3.71375658431355539997467716583, −2.84206545343609058705195086989, −2.15686340953137112061921451226, −1.19915912930715292553423329301, 0,
1.19915912930715292553423329301, 2.15686340953137112061921451226, 2.84206545343609058705195086989, 3.71375658431355539997467716583, 4.28088665839059093669700037659, 4.81103346199117845966800164034, 5.10870910841514021482586673894, 5.65045951154400845042854387589, 6.13709332444057347666756704097, 6.45782929427348210743724639109, 6.89865749107197294369399543033, 7.59776998434374205435069602344, 7.69823812494879928556312846174, 8.604969221559244333634630913098
