| L(s) = 1 | − 2-s − 4-s − 3·7-s + 3·8-s + 4·9-s + 3·14-s − 16-s − 4·17-s − 4·18-s − 2·23-s + 6·25-s + 3·28-s − 12·31-s − 5·32-s + 4·34-s − 4·36-s − 8·41-s + 2·46-s − 6·47-s + 6·49-s − 6·50-s − 9·56-s + 12·62-s − 12·63-s + 7·64-s + 4·68-s − 12·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.13·7-s + 1.06·8-s + 4/3·9-s + 0.801·14-s − 1/4·16-s − 0.970·17-s − 0.942·18-s − 0.417·23-s + 6/5·25-s + 0.566·28-s − 2.15·31-s − 0.883·32-s + 0.685·34-s − 2/3·36-s − 1.24·41-s + 0.294·46-s − 0.875·47-s + 6/7·49-s − 0.848·50-s − 1.20·56-s + 1.52·62-s − 1.51·63-s + 7/8·64-s + 0.485·68-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 257 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 14 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−9.235505516517695740222720913613, −8.846437744261609946052123633423, −8.461099588450724191910510507430, −7.68994689329323238830519046889, −7.24023118377991318787101164367, −6.89858881999266854389289443301, −6.40276588942135201128832155205, −5.67883400118352669751948311086, −4.91395604705488455991646845800, −4.54167292308641693477650631177, −3.77423085784781174805357845419, −3.38475815342893837427756720627, −2.21298773690796926801680029559, −1.38084305691002589034033999586, 0,
1.38084305691002589034033999586, 2.21298773690796926801680029559, 3.38475815342893837427756720627, 3.77423085784781174805357845419, 4.54167292308641693477650631177, 4.91395604705488455991646845800, 5.67883400118352669751948311086, 6.40276588942135201128832155205, 6.89858881999266854389289443301, 7.24023118377991318787101164367, 7.68994689329323238830519046889, 8.461099588450724191910510507430, 8.846437744261609946052123633423, 9.235505516517695740222720913613
