| L(s) = 1 | − 2·2-s + 4-s − 6·5-s + 2·8-s − 3·9-s + 12·10-s − 13-s − 3·16-s + 2·17-s + 6·18-s − 6·19-s − 6·20-s + 19·25-s + 2·26-s − 3·29-s − 6·31-s − 2·32-s − 4·34-s − 3·36-s − 5·37-s + 12·38-s − 12·40-s − 12·41-s + 43-s + 18·45-s + 47-s − 4·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 2.68·5-s + 0.707·8-s − 9-s + 3.79·10-s − 0.277·13-s − 3/4·16-s + 0.485·17-s + 1.41·18-s − 1.37·19-s − 1.34·20-s + 19/5·25-s + 0.392·26-s − 0.557·29-s − 1.07·31-s − 0.353·32-s − 0.685·34-s − 1/2·36-s − 0.821·37-s + 1.94·38-s − 1.89·40-s − 1.87·41-s + 0.152·43-s + 2.68·45-s + 0.145·47-s − 4/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100162 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100162 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 10 T + p T^{2} ) \) |
| 821 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T - 3 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 84 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T - 11 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 96 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 79 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 117 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T - 68 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 99 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−14.7820516310, −14.1978445514, −13.6693529187, −13.0278697170, −12.4333704130, −12.1821311538, −11.8340932066, −11.1884250107, −10.9183832089, −10.6967303665, −10.0140820605, −9.29835429114, −8.99964014164, −8.40568118823, −8.15504904928, −7.90642146099, −7.32009867959, −6.97354916271, −6.20832767531, −5.32092775951, −4.72978866394, −4.11434804284, −3.55285078229, −3.04926181942, −1.67488971209, 0, 0,
1.67488971209, 3.04926181942, 3.55285078229, 4.11434804284, 4.72978866394, 5.32092775951, 6.20832767531, 6.97354916271, 7.32009867959, 7.90642146099, 8.15504904928, 8.40568118823, 8.99964014164, 9.29835429114, 10.0140820605, 10.6967303665, 10.9183832089, 11.1884250107, 11.8340932066, 12.1821311538, 12.4333704130, 13.0278697170, 13.6693529187, 14.1978445514, 14.7820516310
