| 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 + 7·17-s − 3·18-s + 5·19-s − 2·22-s + 2·24-s − 2·25-s − 4·27-s − 32-s − 4·33-s − 7·34-s + 3·36-s − 5·38-s + 17·41-s + 43-s + 2·44-s − 2·48-s + 10·49-s + 2·50-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 + 1.69·17-s − 0.707·18-s + 1.14·19-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 − 1.20·34-s + 1/2·36-s − 0.811·38-s + 2.65·41-s + 0.152·43-s + 0.301·44-s − 0.288·48-s + 10/7·49-s + 0.282·50-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
| \(L(1)\) |
\(\approx\) |
\(1.044614883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.044614883\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
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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
−8.899547731058799229771538395639, −8.143419471126167195769331161019, −7.81703780712628173921183898622, −7.34593913330565877143786586932, −6.99362499590386105194200362515, −6.42781542175905654376957218751, −5.85863707042027363997680434114, −5.47821797150192998903849387985, −5.25929409201904734594585140611, −4.19355607099128123583116261803, −3.94940104277653377776235123622, −3.11034937428825338285960982931, −2.37613023402199732931781555690, −1.28567152849104959722738462143, −0.831414637103149397570797139601,
0.831414637103149397570797139601, 1.28567152849104959722738462143, 2.37613023402199732931781555690, 3.11034937428825338285960982931, 3.94940104277653377776235123622, 4.19355607099128123583116261803, 5.25929409201904734594585140611, 5.47821797150192998903849387985, 5.85863707042027363997680434114, 6.42781542175905654376957218751, 6.99362499590386105194200362515, 7.34593913330565877143786586932, 7.81703780712628173921183898622, 8.143419471126167195769331161019, 8.899547731058799229771538395639
