L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 4·5-s + 4·6-s − 4·8-s − 2·9-s + 8·10-s − 11-s − 6·12-s − 2·13-s + 8·15-s + 5·16-s + 2·17-s + 4·18-s − 4·19-s − 12·20-s + 2·22-s + 4·23-s + 8·24-s + 6·25-s + 4·26-s + 10·27-s − 8·29-s − 16·30-s − 6·31-s − 6·32-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.78·5-s + 1.63·6-s − 1.41·8-s − 2/3·9-s + 2.52·10-s − 0.301·11-s − 1.73·12-s − 0.554·13-s + 2.06·15-s + 5/4·16-s + 0.485·17-s + 0.942·18-s − 0.917·19-s − 2.68·20-s + 0.426·22-s + 0.834·23-s + 1.63·24-s + 6/5·25-s + 0.784·26-s + 1.92·27-s − 1.48·29-s − 2.92·30-s − 1.07·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−18.9755969718, −18.2120541317, −18.2069927431, −17.2128532162, −16.9307018338, −16.3580045805, −16.3370810014, −15.3001198268, −14.9146375694, −14.6077730657, −13.2488477541, −12.3052609901, −12.2206319828, −11.2313614144, −11.1914967506, −10.8242545693, −9.76554711946, −9.02559149679, −8.38389482609, −7.57571100089, −7.41620352139, −6.22872197038, −5.57928681743, −4.35107164129, −2.98342596761, 0,
2.98342596761, 4.35107164129, 5.57928681743, 6.22872197038, 7.41620352139, 7.57571100089, 8.38389482609, 9.02559149679, 9.76554711946, 10.8242545693, 11.1914967506, 11.2313614144, 12.2206319828, 12.3052609901, 13.2488477541, 14.6077730657, 14.9146375694, 15.3001198268, 16.3370810014, 16.3580045805, 16.9307018338, 17.2128532162, 18.2069927431, 18.2120541317, 18.9755969718