| L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·8-s − 5·9-s − 2·10-s − 4·11-s − 13-s − 16-s − 2·17-s − 5·18-s + 8·19-s + 2·20-s − 4·22-s + 4·23-s − 3·25-s − 26-s + 5·32-s − 2·34-s + 5·36-s − 2·37-s + 8·38-s + 6·40-s + 4·44-s + 10·45-s + 4·46-s − 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 5/3·9-s − 0.632·10-s − 1.20·11-s − 0.277·13-s − 1/4·16-s − 0.485·17-s − 1.17·18-s + 1.83·19-s + 0.447·20-s − 0.852·22-s + 0.834·23-s − 3/5·25-s − 0.196·26-s + 0.883·32-s − 0.342·34-s + 5/6·36-s − 0.328·37-s + 1.29·38-s + 0.948·40-s + 0.603·44-s + 1.49·45-s + 0.589·46-s − 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7665631674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7665631674\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.205347839064966795579871824266, −8.862715409353785609204371577456, −8.263830182025717043297208090393, −7.892267268914658419868088037189, −7.60020532881873100679463127988, −6.76748643931637903118557815172, −6.26475998512379981773133490792, −5.40098063242946273847816613023, −5.31317516812660270168242862509, −4.92307280642970823293736146930, −4.01742176703764843798720750283, −3.43156781753479895526053541217, −3.00924040252790436765839620898, −2.37640123005697394146259418847, −0.51513720541533448656868167760,
0.51513720541533448656868167760, 2.37640123005697394146259418847, 3.00924040252790436765839620898, 3.43156781753479895526053541217, 4.01742176703764843798720750283, 4.92307280642970823293736146930, 5.31317516812660270168242862509, 5.40098063242946273847816613023, 6.26475998512379981773133490792, 6.76748643931637903118557815172, 7.60020532881873100679463127988, 7.892267268914658419868088037189, 8.263830182025717043297208090393, 8.862715409353785609204371577456, 9.205347839064966795579871824266
