| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 3·7-s + 8-s + 2·9-s − 2·10-s + 4·11-s + 12-s − 6·13-s + 3·14-s − 2·15-s + 16-s + 2·18-s − 4·19-s − 2·20-s + 3·21-s + 4·22-s + 4·23-s + 24-s + 2·25-s − 6·26-s + 6·27-s + 3·28-s + 5·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 2/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s − 1.66·13-s + 0.801·14-s − 0.516·15-s + 1/4·16-s + 0.471·18-s − 0.917·19-s − 0.447·20-s + 0.654·21-s + 0.852·22-s + 0.834·23-s + 0.204·24-s + 2/5·25-s − 1.17·26-s + 1.15·27-s + 0.566·28-s + 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71408 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71408 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.751116054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.751116054\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 4463 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 54 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T - 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 103 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T - 90 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 67 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 140 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 138 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 168 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 136 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−14.4773514451, −14.0407723923, −13.6540731428, −13.0588973988, −12.3869679371, −12.1896495347, −11.9269843526, −11.3638659375, −10.8252709180, −10.4421819978, −9.82295789474, −9.27785928234, −8.69955192454, −8.16211396274, −7.91297693774, −7.11311735291, −6.88282575201, −6.38123842870, −5.27550326585, −4.77471998001, −4.45428112471, −3.87347082174, −3.05546440396, −2.32332005010, −1.35373366408,
1.35373366408, 2.32332005010, 3.05546440396, 3.87347082174, 4.45428112471, 4.77471998001, 5.27550326585, 6.38123842870, 6.88282575201, 7.11311735291, 7.91297693774, 8.16211396274, 8.69955192454, 9.27785928234, 9.82295789474, 10.4421819978, 10.8252709180, 11.3638659375, 11.9269843526, 12.1896495347, 12.3869679371, 13.0588973988, 13.6540731428, 14.0407723923, 14.4773514451
