L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s + 8-s + 3·10-s − 5·11-s − 3·13-s − 14-s − 3·16-s − 3·20-s + 5·22-s + 9·23-s + 8·25-s + 3·26-s + 28-s − 4·29-s − 31-s + 5·32-s − 3·35-s + 9·37-s − 3·40-s − 4·41-s + 2·43-s − 5·44-s − 9·46-s − 4·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s − 1.50·11-s − 0.832·13-s − 0.267·14-s − 3/4·16-s − 0.670·20-s + 1.06·22-s + 1.87·23-s + 8/5·25-s + 0.588·26-s + 0.188·28-s − 0.742·29-s − 0.179·31-s + 0.883·32-s − 0.507·35-s + 1.47·37-s − 0.474·40-s − 0.624·41-s + 0.304·43-s − 0.753·44-s − 1.32·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 970 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 970 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3475154474\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3475154474\) |
\(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 28 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−19.4386359096, −18.8929149382, −18.5350402938, −17.9690464567, −17.1047375426, −16.7021264024, −16.1605246924, −15.4996253688, −15.0713542355, −14.5976207221, −13.4751902852, −12.9524302057, −12.2948266056, −11.4244789416, −11.0068365658, −10.5128789417, −9.57508637702, −8.72403929332, −7.88618166836, −7.55968001574, −6.86259334572, −5.23325102631, −4.49327541213, −2.85695421779,
2.85695421779, 4.49327541213, 5.23325102631, 6.86259334572, 7.55968001574, 7.88618166836, 8.72403929332, 9.57508637702, 10.5128789417, 11.0068365658, 11.4244789416, 12.2948266056, 12.9524302057, 13.4751902852, 14.5976207221, 15.0713542355, 15.4996253688, 16.1605246924, 16.7021264024, 17.1047375426, 17.9690464567, 18.5350402938, 18.8929149382, 19.4386359096