| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 5·9-s + 14-s + 16-s − 9·17-s − 5·18-s − 2·23-s + 6·25-s + 28-s + 14·31-s + 32-s − 9·34-s − 5·36-s − 2·41-s − 2·46-s + 47-s − 7·49-s + 6·50-s + 56-s + 14·62-s − 5·63-s + 64-s − 9·68-s − 71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 5/3·9-s + 0.267·14-s + 1/4·16-s − 2.18·17-s − 1.17·18-s − 0.417·23-s + 6/5·25-s + 0.188·28-s + 2.51·31-s + 0.176·32-s − 1.54·34-s − 5/6·36-s − 0.312·41-s − 0.294·46-s + 0.145·47-s − 49-s + 0.848·50-s + 0.133·56-s + 1.77·62-s − 0.629·63-s + 1/8·64-s − 1.09·68-s − 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.151473703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.151473703\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−12.08471040033142178812298188935, −11.49774760059071294638775733509, −11.34972120287414887663793048521, −10.63983442207465826752676803199, −10.03678681520190696479507866164, −8.985952130047415582147306772252, −8.598549001061339691920262738964, −8.115194846911258697150048109335, −7.09953951617682447790222367306, −6.37739946488882099217809141615, −5.95471463639550373755376584555, −4.87065807995594601501699640400, −4.49349673887596513143168410470, −3.15129202944543872519784358984, −2.37502736514194649668568101270,
2.37502736514194649668568101270, 3.15129202944543872519784358984, 4.49349673887596513143168410470, 4.87065807995594601501699640400, 5.95471463639550373755376584555, 6.37739946488882099217809141615, 7.09953951617682447790222367306, 8.115194846911258697150048109335, 8.598549001061339691920262738964, 8.985952130047415582147306772252, 10.03678681520190696479507866164, 10.63983442207465826752676803199, 11.34972120287414887663793048521, 11.49774760059071294638775733509, 12.08471040033142178812298188935
