| L(s) = 1 | + 2·3-s − 3·5-s − 3·9-s − 6·15-s − 18·23-s + 4·25-s − 14·27-s + 9·45-s − 24·47-s + 14·49-s + 26·67-s − 36·69-s + 8·75-s − 4·81-s + 18·89-s + 8·103-s + 54·115-s − 11·121-s + 3·125-s + 127-s + 131-s + 42·135-s + 137-s + 139-s − 48·141-s + 28·147-s + 149-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s − 9-s − 1.54·15-s − 3.75·23-s + 4/5·25-s − 2.69·27-s + 1.34·45-s − 3.50·47-s + 2·49-s + 3.17·67-s − 4.33·69-s + 0.923·75-s − 4/9·81-s + 1.90·89-s + 0.788·103-s + 5.03·115-s − 121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 3.61·135-s + 0.0854·137-s + 0.0848·139-s − 4.04·141-s + 2.30·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 774400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 774400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7774620175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7774620175\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−10.25982576111720933788081743823, −9.801650142482725504329898671345, −9.529212944382343090854251976858, −8.933347458556411857058255117055, −8.510768005401465133041672418663, −8.083000695909847026188791092793, −8.030716985202895262372102564848, −7.75652471084123611086287019637, −7.10219615898413498825565471062, −6.35809104130741262893867995270, −6.18413245766100972033696250417, −5.49361782291360259293344026357, −5.08617277414295264632030413508, −4.18631254336666418630552692644, −3.96179033719227057485502362450, −3.47012059363719083948533733764, −3.09752121580266002219979254496, −2.17875658856054625990747197813, −2.03205406972224139897707475422, −0.37032072475410594693555510545,
0.37032072475410594693555510545, 2.03205406972224139897707475422, 2.17875658856054625990747197813, 3.09752121580266002219979254496, 3.47012059363719083948533733764, 3.96179033719227057485502362450, 4.18631254336666418630552692644, 5.08617277414295264632030413508, 5.49361782291360259293344026357, 6.18413245766100972033696250417, 6.35809104130741262893867995270, 7.10219615898413498825565471062, 7.75652471084123611086287019637, 8.030716985202895262372102564848, 8.083000695909847026188791092793, 8.510768005401465133041672418663, 8.933347458556411857058255117055, 9.529212944382343090854251976858, 9.801650142482725504329898671345, 10.25982576111720933788081743823
