| L(s) = 1 | + 2·2-s − 4·3-s + 3·4-s − 4·5-s − 8·6-s − 2·7-s + 4·8-s + 7·9-s − 8·10-s − 2·11-s − 12·12-s − 13-s − 4·14-s + 16·15-s + 5·16-s − 6·17-s + 14·18-s − 12·20-s + 8·21-s − 4·22-s + 2·23-s − 16·24-s + 3·25-s − 2·26-s − 4·27-s − 6·28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 3/2·4-s − 1.78·5-s − 3.26·6-s − 0.755·7-s + 1.41·8-s + 7/3·9-s − 2.52·10-s − 0.603·11-s − 3.46·12-s − 0.277·13-s − 1.06·14-s + 4.13·15-s + 5/4·16-s − 1.45·17-s + 3.29·18-s − 2.68·20-s + 1.74·21-s − 0.852·22-s + 0.417·23-s − 3.26·24-s + 3/5·25-s − 0.392·26-s − 0.769·27-s − 1.13·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−16.7401598333, −16.2105519665, −16.1610422211, −15.4425168943, −15.3407982120, −14.7871828532, −13.5922863880, −13.5230676379, −12.6948580012, −12.1499536241, −12.0906172632, −11.4971267324, −11.1734099077, −10.6571590377, −10.2802576297, −9.14676247888, −8.14577120022, −7.45985560668, −6.89586188103, −6.28038385959, −5.88908450916, −4.97317601948, −4.61153453491, −3.91117219162, −2.87766967615, 0,
2.87766967615, 3.91117219162, 4.61153453491, 4.97317601948, 5.88908450916, 6.28038385959, 6.89586188103, 7.45985560668, 8.14577120022, 9.14676247888, 10.2802576297, 10.6571590377, 11.1734099077, 11.4971267324, 12.0906172632, 12.1499536241, 12.6948580012, 13.5230676379, 13.5922863880, 14.7871828532, 15.3407982120, 15.4425168943, 16.1610422211, 16.2105519665, 16.7401598333
