L(s) = 1 | − 2·3-s + 4-s − 4·5-s + 3·9-s + 11-s − 2·12-s + 8·15-s + 16-s − 4·20-s − 8·23-s + 2·25-s − 4·27-s − 8·31-s − 2·33-s + 3·36-s − 4·37-s + 44-s − 12·45-s − 16·47-s − 2·48-s + 49-s − 28·53-s − 4·55-s + 24·59-s + 8·60-s + 64-s + 8·67-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/2·4-s − 1.78·5-s + 9-s + 0.301·11-s − 0.577·12-s + 2.06·15-s + 1/4·16-s − 0.894·20-s − 1.66·23-s + 2/5·25-s − 0.769·27-s − 1.43·31-s − 0.348·33-s + 1/2·36-s − 0.657·37-s + 0.150·44-s − 1.78·45-s − 2.33·47-s − 0.288·48-s + 1/7·49-s − 3.84·53-s − 0.539·55-s + 3.12·59-s + 1.03·60-s + 1/8·64-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2347884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.13188734812014929646560473693, −6.91394332771191655575545668718, −6.36362927444081124983699082758, −6.17569848665337782312463950999, −5.47060768957514705173421460731, −5.12658951935756468823062559161, −4.72832473817225769292057457440, −3.95586600708170815846951891453, −3.72716170584453900058590617219, −3.62319053698876325772393238292, −2.60592425173918161360361657651, −1.87596574858564202101556695192, −1.31751202956692819023526147888, 0, 0,
1.31751202956692819023526147888, 1.87596574858564202101556695192, 2.60592425173918161360361657651, 3.62319053698876325772393238292, 3.72716170584453900058590617219, 3.95586600708170815846951891453, 4.72832473817225769292057457440, 5.12658951935756468823062559161, 5.47060768957514705173421460731, 6.17569848665337782312463950999, 6.36362927444081124983699082758, 6.91394332771191655575545668718, 7.13188734812014929646560473693