| L(s) = 1 | − 2·2-s − 2·3-s + 4-s − 3·5-s + 4·6-s − 7-s + 2·9-s + 6·10-s − 7·11-s − 2·12-s + 4·13-s + 2·14-s + 6·15-s + 16-s + 4·17-s − 4·18-s − 3·19-s − 3·20-s + 2·21-s + 14·22-s − 3·23-s + 2·25-s − 8·26-s − 6·27-s − 28-s + 4·29-s − 12·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s + 1.63·6-s − 0.377·7-s + 2/3·9-s + 1.89·10-s − 2.11·11-s − 0.577·12-s + 1.10·13-s + 0.534·14-s + 1.54·15-s + 1/4·16-s + 0.970·17-s − 0.942·18-s − 0.688·19-s − 0.670·20-s + 0.436·21-s + 2.98·22-s − 0.625·23-s + 2/5·25-s − 1.56·26-s − 1.15·27-s − 0.188·28-s + 0.742·29-s − 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1499 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1499 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 1499 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 36 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 3 p T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 83 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 166 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7 T + 175 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−19.2652029943, −18.9056954065, −18.4044920760, −18.0020772672, −17.7086914145, −16.7364213550, −16.5713789365, −15.7634625060, −15.6186952622, −15.0508845390, −13.8017601704, −13.2385638377, −12.4241209590, −12.0816511382, −11.1837819079, −10.8696964455, −10.1301173928, −9.69388579168, −8.55923625840, −7.98041829618, −7.78090172012, −6.51655485872, −5.71765195660, −4.72634893110, −3.40759643562, 0,
3.40759643562, 4.72634893110, 5.71765195660, 6.51655485872, 7.78090172012, 7.98041829618, 8.55923625840, 9.69388579168, 10.1301173928, 10.8696964455, 11.1837819079, 12.0816511382, 12.4241209590, 13.2385638377, 13.8017601704, 15.0508845390, 15.6186952622, 15.7634625060, 16.5713789365, 16.7364213550, 17.7086914145, 18.0020772672, 18.4044920760, 18.9056954065, 19.2652029943
