| L(s) = 1 | + 2-s − 4-s − 2·7-s − 8-s − 3·9-s − 3·13-s − 2·14-s + 3·16-s + 3·17-s − 3·18-s + 4·19-s + 3·23-s − 4·25-s − 3·26-s + 2·28-s + 6·29-s − 8·31-s + 3·32-s + 3·34-s + 3·36-s − 2·37-s + 4·38-s + 43-s + 3·46-s + 6·47-s − 2·49-s − 4·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.755·7-s − 0.353·8-s − 9-s − 0.832·13-s − 0.534·14-s + 3/4·16-s + 0.727·17-s − 0.707·18-s + 0.917·19-s + 0.625·23-s − 4/5·25-s − 0.588·26-s + 0.377·28-s + 1.11·29-s − 1.43·31-s + 0.530·32-s + 0.514·34-s + 1/2·36-s − 0.328·37-s + 0.648·38-s + 0.152·43-s + 0.442·46-s + 0.875·47-s − 2/7·49-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7370872562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7370872562\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−18.8309656382, −18.3009925239, −17.6038071339, −17.0701016621, −16.7999068210, −15.9734454274, −15.5589974382, −14.6433530531, −14.3642331228, −13.9052612993, −13.3145275460, −12.6296340044, −12.2707653305, −11.6206444984, −10.8424849460, −10.0332484235, −9.51985634328, −8.91045540801, −7.99439336676, −7.37033721363, −6.33293288284, −5.49155343444, −4.99960065850, −3.75482663883, −2.95329093402,
2.95329093402, 3.75482663883, 4.99960065850, 5.49155343444, 6.33293288284, 7.37033721363, 7.99439336676, 8.91045540801, 9.51985634328, 10.0332484235, 10.8424849460, 11.6206444984, 12.2707653305, 12.6296340044, 13.3145275460, 13.9052612993, 14.3642331228, 14.6433530531, 15.5589974382, 15.9734454274, 16.7999068210, 17.0701016621, 17.6038071339, 18.3009925239, 18.8309656382
