| L(s) = 1 | − 2-s + 3-s − 6-s − 3·7-s − 8-s − 2·9-s + 11-s + 13-s + 3·14-s − 16-s + 7·17-s + 2·18-s − 19-s − 3·21-s − 22-s − 3·23-s − 24-s − 25-s − 26-s − 2·27-s + 6·29-s − 6·31-s + 6·32-s + 33-s − 7·34-s + 4·37-s + 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.801·14-s − 1/4·16-s + 1.69·17-s + 0.471·18-s − 0.229·19-s − 0.654·21-s − 0.213·22-s − 0.625·23-s − 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.384·27-s + 1.11·29-s − 1.07·31-s + 1.06·32-s + 0.174·33-s − 1.20·34-s + 0.657·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1253 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1253 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4149278700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4149278700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 179 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 13 T + 115 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 86 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 145 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T - 38 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−19.6278259220, −19.0208900300, −18.5858261920, −18.1222413739, −17.4524490769, −16.8036468937, −16.3743885478, −15.8744533187, −15.0496601513, −14.5868805607, −13.9189615839, −13.4282007917, −12.6261876085, −12.0071559610, −11.4655210862, −10.4000574603, −9.80282695730, −9.39898966712, −8.49680800428, −8.24807134334, −7.07969002645, −6.28859984865, −5.45639773760, −3.79810289160, −2.86998844307,
2.86998844307, 3.79810289160, 5.45639773760, 6.28859984865, 7.07969002645, 8.24807134334, 8.49680800428, 9.39898966712, 9.80282695730, 10.4000574603, 11.4655210862, 12.0071559610, 12.6261876085, 13.4282007917, 13.9189615839, 14.5868805607, 15.0496601513, 15.8744533187, 16.3743885478, 16.8036468937, 17.4524490769, 18.1222413739, 18.5858261920, 19.0208900300, 19.6278259220
