L(s) = 1 | + 2-s − 4-s + 2·5-s − 3·8-s − 6·9-s + 2·10-s + 4·13-s − 16-s + 12·17-s − 6·18-s − 2·20-s + 3·25-s + 4·26-s + 12·29-s + 5·32-s + 12·34-s + 6·36-s − 4·37-s − 6·40-s + 4·41-s − 12·45-s − 14·49-s + 3·50-s − 4·52-s − 4·53-s + 12·58-s − 20·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s − 2·9-s + 0.632·10-s + 1.10·13-s − 1/4·16-s + 2.91·17-s − 1.41·18-s − 0.447·20-s + 3/5·25-s + 0.784·26-s + 2.22·29-s + 0.883·32-s + 2.05·34-s + 36-s − 0.657·37-s − 0.948·40-s + 0.624·41-s − 1.78·45-s − 2·49-s + 0.424·50-s − 0.554·52-s − 0.549·53-s + 1.57·58-s − 2.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.770622110\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.770622110\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−10.07588915028781859505118646541, −9.376036497146997809466572379928, −9.365267708430525132762162914792, −8.402071812243779349662792300606, −8.248642768475844318802620430882, −7.78041223670494609407513138154, −6.49908708730306794178412805091, −6.16821463973303244938343944630, −5.83555924118533390298503875284, −5.09766059609940450216711100862, −4.97617253141282559040022801022, −3.60799086718076388676488423337, −3.25423179226253980082861279896, −2.69303075235727048947048138558, −1.13555459814448309624758433433,
1.13555459814448309624758433433, 2.69303075235727048947048138558, 3.25423179226253980082861279896, 3.60799086718076388676488423337, 4.97617253141282559040022801022, 5.09766059609940450216711100862, 5.83555924118533390298503875284, 6.16821463973303244938343944630, 6.49908708730306794178412805091, 7.78041223670494609407513138154, 8.248642768475844318802620430882, 8.402071812243779349662792300606, 9.365267708430525132762162914792, 9.376036497146997809466572379928, 10.07588915028781859505118646541