| L(s) = 1 | − 3-s + 4-s + 3·5-s − 3·9-s + 3·11-s − 12-s − 3·15-s + 16-s + 3·20-s − 23-s − 25-s + 4·27-s + 31-s − 3·33-s − 3·36-s − 5·37-s + 3·44-s − 9·45-s − 3·47-s − 48-s + 5·49-s + 9·53-s + 9·55-s − 15·59-s − 3·60-s + 64-s + 10·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s + 1.34·5-s − 9-s + 0.904·11-s − 0.288·12-s − 0.774·15-s + 1/4·16-s + 0.670·20-s − 0.208·23-s − 1/5·25-s + 0.769·27-s + 0.179·31-s − 0.522·33-s − 1/2·36-s − 0.821·37-s + 0.452·44-s − 1.34·45-s − 0.437·47-s − 0.144·48-s + 5/7·49-s + 1.23·53-s + 1.21·55-s − 1.95·59-s − 0.387·60-s + 1/8·64-s + 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.104030657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104030657\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−11.56316394653376570244995234947, −10.81462360811959856626171665076, −10.37012759895516523672648436360, −9.861078755168893808314804524080, −9.120944525617901054294188021261, −8.842200867151660923052271879475, −7.974177590350845099527593512310, −7.21914308958190089406562875644, −6.46201834386554581756027099050, −6.00793682321193651645432528267, −5.66198702338098475826250876804, −4.85763786525907856945100917909, −3.74185835322883559775818805113, −2.69618367775349164641980059698, −1.70899833148032017751541103830,
1.70899833148032017751541103830, 2.69618367775349164641980059698, 3.74185835322883559775818805113, 4.85763786525907856945100917909, 5.66198702338098475826250876804, 6.00793682321193651645432528267, 6.46201834386554581756027099050, 7.21914308958190089406562875644, 7.974177590350845099527593512310, 8.842200867151660923052271879475, 9.120944525617901054294188021261, 9.861078755168893808314804524080, 10.37012759895516523672648436360, 10.81462360811959856626171665076, 11.56316394653376570244995234947
