| L(s) = 1 | − 2·2-s − 4·3-s + 3·4-s + 8·6-s + 2·7-s − 4·8-s + 6·9-s − 12·12-s − 8·13-s − 4·14-s + 5·16-s + 12·17-s − 12·18-s + 4·19-s − 8·21-s + 16·24-s − 10·25-s + 16·26-s + 4·27-s + 6·28-s − 12·29-s − 8·31-s − 6·32-s − 24·34-s + 18·36-s + 4·37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 3/2·4-s + 3.26·6-s + 0.755·7-s − 1.41·8-s + 2·9-s − 3.46·12-s − 2.21·13-s − 1.06·14-s + 5/4·16-s + 2.91·17-s − 2.82·18-s + 0.917·19-s − 1.74·21-s + 3.26·24-s − 2·25-s + 3.13·26-s + 0.769·27-s + 1.13·28-s − 2.22·29-s − 1.43·31-s − 1.06·32-s − 4.11·34-s + 3·36-s + 0.657·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1090476651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1090476651\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−19.5800270198, −19.5800270198, −18.2120541317, −18.2120541317, −17.2128532162, −17.2128532162, −16.3370810014, −16.3370810014, −14.6077730657, −14.6077730657, −12.3052609901, −12.3052609901, −11.2313614144, −11.2313614144, −9.76554711946, −9.76554711946, −7.57571100089, −7.57571100089, −5.57928681743, −5.57928681743,
5.57928681743, 5.57928681743, 7.57571100089, 7.57571100089, 9.76554711946, 9.76554711946, 11.2313614144, 11.2313614144, 12.3052609901, 12.3052609901, 14.6077730657, 14.6077730657, 16.3370810014, 16.3370810014, 17.2128532162, 17.2128532162, 18.2120541317, 18.2120541317, 19.5800270198, 19.5800270198
