| L(s) = 1 | − 4·2-s + 8·4-s − 3·7-s − 8·8-s − 4·11-s + 12·14-s − 4·16-s + 16·22-s + 12·23-s + 25-s − 24·28-s + 4·29-s + 32·32-s + 10·37-s + 8·43-s − 32·44-s − 48·46-s + 2·49-s − 4·50-s − 4·53-s + 24·56-s − 16·58-s − 64·64-s − 18·67-s + 4·71-s − 40·74-s + 12·77-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 1.13·7-s − 2.82·8-s − 1.20·11-s + 3.20·14-s − 16-s + 3.41·22-s + 2.50·23-s + 1/5·25-s − 4.53·28-s + 0.742·29-s + 5.65·32-s + 1.64·37-s + 1.21·43-s − 4.82·44-s − 7.07·46-s + 2/7·49-s − 0.565·50-s − 0.549·53-s + 3.20·56-s − 2.10·58-s − 8·64-s − 2.19·67-s + 0.474·71-s − 4.64·74-s + 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 893025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−7.899783450978500873942229777226, −7.72779393764142968328912399540, −7.44584033135863585609716971257, −6.75609439138463161761765040790, −6.68799725367635564271959497809, −6.01927549672502062107062715178, −5.36088394579301731865192759191, −4.69221690938344723805408149899, −4.40572769350290061562549596313, −3.34052893736309811768832391122, −2.65254993398592503147695516254, −2.53645300939543118255952249267, −1.36907081500796754283112846338, −0.840733435330795574728362410434, 0,
0.840733435330795574728362410434, 1.36907081500796754283112846338, 2.53645300939543118255952249267, 2.65254993398592503147695516254, 3.34052893736309811768832391122, 4.40572769350290061562549596313, 4.69221690938344723805408149899, 5.36088394579301731865192759191, 6.01927549672502062107062715178, 6.68799725367635564271959497809, 6.75609439138463161761765040790, 7.44584033135863585609716971257, 7.72779393764142968328912399540, 7.899783450978500873942229777226
