L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 4·9-s − 7·11-s − 2·12-s + 16-s − 6·17-s − 4·18-s − 4·19-s + 7·22-s + 2·24-s + 4·25-s − 5·27-s − 32-s + 14·33-s + 6·34-s + 4·36-s + 4·38-s − 11·41-s − 43-s − 7·44-s − 2·48-s − 11·49-s − 4·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 4/3·9-s − 2.11·11-s − 0.577·12-s + 1/4·16-s − 1.45·17-s − 0.942·18-s − 0.917·19-s + 1.49·22-s + 0.408·24-s + 4/5·25-s − 0.962·27-s − 0.176·32-s + 2.43·33-s + 1.02·34-s + 2/3·36-s + 0.648·38-s − 1.71·41-s − 0.152·43-s − 1.05·44-s − 0.288·48-s − 1.57·49-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 12 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.84029278903557493797460719162, −10.46750903863308980307392268619, −9.894091456753804767688149230613, −9.304367627546415876073709126418, −8.399503040276540200170274749395, −8.187649934932811790541702864723, −7.35993356036733808755043340184, −6.71652536738266069584465467093, −6.42162768793371288659243844033, −5.42732003445070923307315686624, −4.99489014438729461108704082071, −4.29854184095109999611534361966, −2.95528667143029691869062466521, −1.94877540993939406225847659283, 0,
1.94877540993939406225847659283, 2.95528667143029691869062466521, 4.29854184095109999611534361966, 4.99489014438729461108704082071, 5.42732003445070923307315686624, 6.42162768793371288659243844033, 6.71652536738266069584465467093, 7.35993356036733808755043340184, 8.187649934932811790541702864723, 8.399503040276540200170274749395, 9.304367627546415876073709126418, 9.894091456753804767688149230613, 10.46750903863308980307392268619, 10.84029278903557493797460719162