| L(s) = 1 | − 2·2-s − 4·3-s + 4-s − 3·5-s + 8·6-s − 7-s + 6·9-s + 6·10-s − 6·11-s − 4·12-s − 3·13-s + 2·14-s + 12·15-s + 16-s − 4·17-s − 12·18-s − 19-s − 3·20-s + 4·21-s + 12·22-s − 7·23-s + 8·25-s + 6·26-s + 4·27-s − 28-s + 6·29-s − 24·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s − 1.34·5-s + 3.26·6-s − 0.377·7-s + 2·9-s + 1.89·10-s − 1.80·11-s − 1.15·12-s − 0.832·13-s + 0.534·14-s + 3.09·15-s + 1/4·16-s − 0.970·17-s − 2.82·18-s − 0.229·19-s − 0.670·20-s + 0.872·21-s + 2.55·22-s − 1.45·23-s + 8/5·25-s + 1.17·26-s + 0.769·27-s − 0.188·28-s + 1.11·29-s − 4.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7165 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7165 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 1433 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 34 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 86 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 2 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 92 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 26 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 240 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−17.5954652855, −17.1618300644, −16.6734506598, −16.1586599489, −15.7856712111, −15.6050005040, −14.7632187118, −13.9930474232, −13.1268514144, −12.5458112695, −12.1198832533, −11.7551998036, −11.2403384152, −10.5464085999, −10.3961483092, −9.88170401658, −8.75861753340, −8.43271993934, −7.82179147315, −7.04015264766, −6.49987538805, −5.67579823181, −5.00604408141, −4.47379472706, −2.92391090258, 0, 0,
2.92391090258, 4.47379472706, 5.00604408141, 5.67579823181, 6.49987538805, 7.04015264766, 7.82179147315, 8.43271993934, 8.75861753340, 9.88170401658, 10.3961483092, 10.5464085999, 11.2403384152, 11.7551998036, 12.1198832533, 12.5458112695, 13.1268514144, 13.9930474232, 14.7632187118, 15.6050005040, 15.7856712111, 16.1586599489, 16.6734506598, 17.1618300644, 17.5954652855
