| L(s) = 1 | − 2-s − 4·3-s + 4-s + 4·6-s − 3·7-s − 8-s + 7·9-s − 2·11-s − 4·12-s − 13-s + 3·14-s + 16-s − 4·17-s − 7·18-s − 5·19-s + 12·21-s + 2·22-s − 2·23-s + 4·24-s − 5·25-s + 26-s − 4·27-s − 3·28-s − 3·29-s − 6·31-s − 32-s + 8·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s + 1.63·6-s − 1.13·7-s − 0.353·8-s + 7/3·9-s − 0.603·11-s − 1.15·12-s − 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.970·17-s − 1.64·18-s − 1.14·19-s + 2.61·21-s + 0.426·22-s − 0.417·23-s + 0.816·24-s − 25-s + 0.196·26-s − 0.769·27-s − 0.566·28-s − 0.557·29-s − 1.07·31-s − 0.176·32-s + 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88768 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88768 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 47 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T - 37 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T - 55 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 96 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 83 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 17 T + 225 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−14.6779148696, −14.3173998535, −13.3139992402, −13.0388521599, −12.8921127736, −12.1846993911, −11.8184517782, −11.4867017691, −11.0461809695, −10.5351462760, −10.3966975867, −9.79995807233, −9.32032301483, −8.77385046172, −8.16890501989, −7.44691081223, −6.99990788364, −6.45853579023, −6.04675250257, −5.79292809105, −5.12631814920, −4.50011588670, −3.75443029100, −2.75393150577, −1.79211248272, 0, 0,
1.79211248272, 2.75393150577, 3.75443029100, 4.50011588670, 5.12631814920, 5.79292809105, 6.04675250257, 6.45853579023, 6.99990788364, 7.44691081223, 8.16890501989, 8.77385046172, 9.32032301483, 9.79995807233, 10.3966975867, 10.5351462760, 11.0461809695, 11.4867017691, 11.8184517782, 12.1846993911, 12.8921127736, 13.0388521599, 13.3139992402, 14.3173998535, 14.6779148696
