| L(s) = 1 | + 2-s + 2·3-s − 4-s − 3·5-s + 2·6-s + 7-s − 3·8-s + 3·9-s − 3·10-s − 4·11-s − 2·12-s − 13-s + 14-s − 6·15-s − 16-s + 4·17-s + 3·18-s + 8·19-s + 3·20-s + 2·21-s − 4·22-s + 16·23-s − 6·24-s + 2·25-s − 26-s + 4·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 1.34·5-s + 0.816·6-s + 0.377·7-s − 1.06·8-s + 9-s − 0.948·10-s − 1.20·11-s − 0.577·12-s − 0.277·13-s + 0.267·14-s − 1.54·15-s − 1/4·16-s + 0.970·17-s + 0.707·18-s + 1.83·19-s + 0.670·20-s + 0.436·21-s − 0.852·22-s + 3.33·23-s − 1.22·24-s + 2/5·25-s − 0.196·26-s + 0.769·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.064067671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064067671\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| good | 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−14.5173856628, −14.1457771429, −13.4092975284, −13.3565767132, −12.7655567554, −12.3221586340, −12.0056456898, −11.3133741264, −11.0205906871, −10.3331970762, −9.65679775515, −9.40319235440, −8.80018193273, −8.17328234983, −8.03084901224, −7.26958585489, −7.23869976315, −6.17024802828, −5.11340090096, −5.01357758336, −4.53457532458, −3.48385429021, −3.14826068230, −2.85446571167, −1.07211392054,
1.07211392054, 2.85446571167, 3.14826068230, 3.48385429021, 4.53457532458, 5.01357758336, 5.11340090096, 6.17024802828, 7.23869976315, 7.26958585489, 8.03084901224, 8.17328234983, 8.80018193273, 9.40319235440, 9.65679775515, 10.3331970762, 11.0205906871, 11.3133741264, 12.0056456898, 12.3221586340, 12.7655567554, 13.3565767132, 13.4092975284, 14.1457771429, 14.5173856628
