| L(s) = 1 | − 2·2-s − 4·3-s − 6·5-s + 8·6-s − 4·7-s + 4·8-s + 6·9-s + 12·10-s − 2·11-s − 8·13-s + 8·14-s + 24·15-s − 4·16-s − 6·17-s − 12·18-s − 3·19-s + 16·21-s + 4·22-s + 2·23-s − 16·24-s + 18·25-s + 16·26-s + 4·27-s − 10·29-s − 48·30-s − 4·31-s + 8·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s − 2.68·5-s + 3.26·6-s − 1.51·7-s + 1.41·8-s + 2·9-s + 3.79·10-s − 0.603·11-s − 2.21·13-s + 2.13·14-s + 6.19·15-s − 16-s − 1.45·17-s − 2.82·18-s − 0.688·19-s + 3.49·21-s + 0.852·22-s + 0.417·23-s − 3.26·24-s + 18/5·25-s + 3.13·26-s + 0.769·27-s − 1.85·29-s − 8.76·30-s − 0.718·31-s + 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35131 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35131 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
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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
−15.9802273957, −15.6836238883, −15.2173166248, −14.7879696813, −13.9683938385, −13.0385428251, −12.6877656582, −12.3834323879, −11.9939812333, −11.5135334923, −10.9431905151, −10.7971801876, −10.3253943857, −9.51289740609, −9.20679535768, −8.60608543705, −7.86442320164, −7.57518925878, −6.82871744579, −6.74965079589, −5.62127290757, −5.03021166776, −4.49472027398, −3.96877979243, −2.89569102446, 0, 0, 0,
2.89569102446, 3.96877979243, 4.49472027398, 5.03021166776, 5.62127290757, 6.74965079589, 6.82871744579, 7.57518925878, 7.86442320164, 8.60608543705, 9.20679535768, 9.51289740609, 10.3253943857, 10.7971801876, 10.9431905151, 11.5135334923, 11.9939812333, 12.3834323879, 12.6877656582, 13.0385428251, 13.9683938385, 14.7879696813, 15.2173166248, 15.6836238883, 15.9802273957
