| L(s) = 1 | − 5-s + 9-s − 5·13-s − 10·17-s − 4·25-s + 5·29-s − 5·37-s − 14·41-s − 45-s + 5·49-s − 10·53-s + 11·61-s + 5·65-s + 81-s + 10·85-s + 5·89-s + 9·101-s + 10·113-s − 5·117-s − 8·121-s + 9·125-s + 127-s + 131-s + 137-s + 139-s − 5·145-s + 149-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1/3·9-s − 1.38·13-s − 2.42·17-s − 4/5·25-s + 0.928·29-s − 0.821·37-s − 2.18·41-s − 0.149·45-s + 5/7·49-s − 1.37·53-s + 1.40·61-s + 0.620·65-s + 1/9·81-s + 1.08·85-s + 0.529·89-s + 0.895·101-s + 0.940·113-s − 0.462·117-s − 0.727·121-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.415·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.787541453195379106337955914492, −9.210793669823180863792099826138, −8.653727689453629661452713032755, −8.322518280853064143024619305345, −7.59220440248120262588618711954, −7.10893839505242258298399591442, −6.67178555609508559106138033204, −6.20832243743736486178727665123, −5.14780180772322744013064449950, −4.83344762762621157989972660940, −4.23016655512540778361992955493, −3.53144890925837273552987151291, −2.54787905031139212293900406244, −1.90748335610678445424063891107, 0,
1.90748335610678445424063891107, 2.54787905031139212293900406244, 3.53144890925837273552987151291, 4.23016655512540778361992955493, 4.83344762762621157989972660940, 5.14780180772322744013064449950, 6.20832243743736486178727665123, 6.67178555609508559106138033204, 7.10893839505242258298399591442, 7.59220440248120262588618711954, 8.322518280853064143024619305345, 8.653727689453629661452713032755, 9.210793669823180863792099826138, 9.787541453195379106337955914492
