L(s) = 1 | − 3-s + 9-s − 4·11-s − 4·17-s − 6·25-s − 27-s − 12·29-s + 16·31-s + 4·33-s + 12·37-s + 12·41-s − 14·49-s + 4·51-s − 8·67-s + 6·75-s + 81-s + 8·83-s + 12·87-s − 16·93-s + 4·97-s − 4·99-s + 36·101-s + 32·103-s + 24·107-s − 12·111-s + 5·121-s − 12·123-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.970·17-s − 6/5·25-s − 0.192·27-s − 2.22·29-s + 2.87·31-s + 0.696·33-s + 1.97·37-s + 1.87·41-s − 2·49-s + 0.560·51-s − 0.977·67-s + 0.692·75-s + 1/9·81-s + 0.878·83-s + 1.28·87-s − 1.65·93-s + 0.406·97-s − 0.402·99-s + 3.58·101-s + 3.15·103-s + 2.32·107-s − 1.13·111-s + 5/11·121-s − 1.08·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9893718403\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9893718403\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.065170023749915638598404079220, −8.632814053171275790838666005226, −7.88365108150257423936793116264, −7.62403207119791885504595168937, −7.41145043458246524552596785021, −6.34034973016640940830304663850, −6.14969218484336320739052245813, −5.85561327512449935358731086305, −4.83326855247570334999188527992, −4.76750339316057597575339371032, −4.08582339913610742428343396917, −3.30728356810877078064881030636, −2.51659494204614894722398853037, −1.97854038896010733538639767531, −0.63207175719826203780734613127,
0.63207175719826203780734613127, 1.97854038896010733538639767531, 2.51659494204614894722398853037, 3.30728356810877078064881030636, 4.08582339913610742428343396917, 4.76750339316057597575339371032, 4.83326855247570334999188527992, 5.85561327512449935358731086305, 6.14969218484336320739052245813, 6.34034973016640940830304663850, 7.41145043458246524552596785021, 7.62403207119791885504595168937, 7.88365108150257423936793116264, 8.632814053171275790838666005226, 9.065170023749915638598404079220