| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 2·12-s + 16-s + 18-s + 6·19-s − 2·24-s + 4·25-s + 4·27-s − 4·29-s + 32-s + 36-s + 6·38-s − 8·41-s + 12·43-s − 2·48-s + 2·49-s + 4·50-s + 20·53-s + 4·54-s − 12·57-s − 4·58-s − 12·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.577·12-s + 1/4·16-s + 0.235·18-s + 1.37·19-s − 0.408·24-s + 4/5·25-s + 0.769·27-s − 0.742·29-s + 0.176·32-s + 1/6·36-s + 0.973·38-s − 1.24·41-s + 1.82·43-s − 0.288·48-s + 2/7·49-s + 0.565·50-s + 2.74·53-s + 0.544·54-s − 1.58·57-s − 0.525·58-s − 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437366681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437366681\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
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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
−10.10978039462609460569329147385, −9.832052367517959198864568615049, −8.993079421979404553843135518520, −8.636959924285806862957815444303, −7.75215155129075519691970554036, −7.30263532073681822887175737347, −6.81450755651330562716835051096, −6.22407519748102899094010192929, −5.55949286936483349312318213484, −5.35664320055366350688400020972, −4.70584126600704787648785915360, −3.96375575355435009569611263424, −3.23647961879955749418780450259, −2.38020445127025878992839105854, −1.03331899838349066910492890113,
1.03331899838349066910492890113, 2.38020445127025878992839105854, 3.23647961879955749418780450259, 3.96375575355435009569611263424, 4.70584126600704787648785915360, 5.35664320055366350688400020972, 5.55949286936483349312318213484, 6.22407519748102899094010192929, 6.81450755651330562716835051096, 7.30263532073681822887175737347, 7.75215155129075519691970554036, 8.636959924285806862957815444303, 8.993079421979404553843135518520, 9.832052367517959198864568615049, 10.10978039462609460569329147385
