| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s − 2·9-s + 12-s + 16-s − 2·18-s + 13·19-s + 24-s − 10·25-s − 5·27-s + 3·29-s + 32-s − 2·36-s + 13·38-s − 2·43-s + 15·47-s + 48-s + 5·49-s − 10·50-s − 3·53-s − 5·54-s + 13·57-s + 3·58-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.288·12-s + 1/4·16-s − 0.471·18-s + 2.98·19-s + 0.204·24-s − 2·25-s − 0.962·27-s + 0.557·29-s + 0.176·32-s − 1/3·36-s + 2.10·38-s − 0.304·43-s + 2.18·47-s + 0.144·48-s + 5/7·49-s − 1.41·50-s − 0.412·53-s − 0.680·54-s + 1.72·57-s + 0.393·58-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 551808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 551808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.755702238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.755702238\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 479 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 15 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−8.192729030215221538252190338655, −7.991416115547882357600279005987, −7.63626779309136085647225901992, −7.08907387468072677905855032780, −6.73593911081944732369335280952, −5.90348834806764702901103154331, −5.59000963688081112867670227879, −5.39311546178493159212924186801, −4.70316076621571339256658656490, −3.90727913050279887195485811418, −3.68235642402039979929937519370, −3.01323735089998690933265752510, −2.59129701008556045456863576736, −1.86797539276807204989382430811, −0.891432368591028076510194418065,
0.891432368591028076510194418065, 1.86797539276807204989382430811, 2.59129701008556045456863576736, 3.01323735089998690933265752510, 3.68235642402039979929937519370, 3.90727913050279887195485811418, 4.70316076621571339256658656490, 5.39311546178493159212924186801, 5.59000963688081112867670227879, 5.90348834806764702901103154331, 6.73593911081944732369335280952, 7.08907387468072677905855032780, 7.63626779309136085647225901992, 7.991416115547882357600279005987, 8.192729030215221538252190338655
