| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s + 3·11-s + 12-s + 4·13-s − 16-s − 18-s − 3·22-s + 8·23-s − 3·24-s − 6·25-s − 4·26-s − 27-s − 5·32-s − 3·33-s − 36-s − 4·37-s − 4·39-s − 3·44-s − 8·46-s − 8·47-s + 48-s + 2·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 1.10·13-s − 1/4·16-s − 0.235·18-s − 0.639·22-s + 1.66·23-s − 0.612·24-s − 6/5·25-s − 0.784·26-s − 0.192·27-s − 0.883·32-s − 0.522·33-s − 1/6·36-s − 0.657·37-s − 0.640·39-s − 0.452·44-s − 1.17·46-s − 1.16·47-s + 0.144·48-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4802185861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4802185861\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−12.22929985017712567797235022800, −11.55281903357586530802457408363, −11.06554417879310638446300217435, −10.59301782729394958636887779974, −9.842386745396970440955645110138, −9.338826184149724184439092868839, −8.770564854925164249035369230573, −8.228260735278396584280170930047, −7.39390228197282041656778383503, −6.74603138048678310026121780637, −5.98467088948894721984616499875, −5.15868189575557544724239962001, −4.31393854123724027348040128745, −3.49049287814637479666504094012, −1.41992576303479051574990026607,
1.41992576303479051574990026607, 3.49049287814637479666504094012, 4.31393854123724027348040128745, 5.15868189575557544724239962001, 5.98467088948894721984616499875, 6.74603138048678310026121780637, 7.39390228197282041656778383503, 8.228260735278396584280170930047, 8.770564854925164249035369230573, 9.338826184149724184439092868839, 9.842386745396970440955645110138, 10.59301782729394958636887779974, 11.06554417879310638446300217435, 11.55281903357586530802457408363, 12.22929985017712567797235022800
