| L(s) = 1 | + 4·5-s − 4·11-s + 16·23-s + 2·25-s + 16·31-s + 12·37-s − 14·49-s + 4·53-s − 16·55-s − 8·59-s − 8·67-s − 16·71-s + 12·89-s + 4·97-s + 32·103-s − 36·113-s + 64·115-s + 5·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 64·155-s + 157-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.20·11-s + 3.33·23-s + 2/5·25-s + 2.87·31-s + 1.97·37-s − 2·49-s + 0.549·53-s − 2.15·55-s − 1.04·59-s − 0.977·67-s − 1.89·71-s + 1.27·89-s + 0.406·97-s + 3.15·103-s − 3.38·113-s + 5.96·115-s + 5/11·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 5.14·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.922928979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.922928979\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.412059248281827571082850459164, −7.83654723068629114933839597949, −7.62403207119791885504595168937, −6.92796334648786182474479663711, −6.40725479078387541676599821855, −6.14969218484336320739052245813, −5.70681982833013824339552087978, −5.05736853539339825666279063330, −4.83326855247570334999188527992, −4.38461660538881471089251752313, −3.21029387756548432515894957212, −2.80691305312846013255691010566, −2.51659494204614894722398853037, −1.61011393533291281462985460586, −0.918192672118257559483050015003,
0.918192672118257559483050015003, 1.61011393533291281462985460586, 2.51659494204614894722398853037, 2.80691305312846013255691010566, 3.21029387756548432515894957212, 4.38461660538881471089251752313, 4.83326855247570334999188527992, 5.05736853539339825666279063330, 5.70681982833013824339552087978, 6.14969218484336320739052245813, 6.40725479078387541676599821855, 6.92796334648786182474479663711, 7.62403207119791885504595168937, 7.83654723068629114933839597949, 8.412059248281827571082850459164
