| L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 2·10-s − 4·16-s − 2·20-s + 25-s + 4·31-s − 8·32-s + 2·49-s + 2·50-s + 20·53-s + 8·62-s − 8·64-s − 4·79-s + 4·80-s + 4·98-s + 2·100-s + 40·106-s + 40·107-s − 18·121-s + 8·124-s − 125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.632·10-s − 16-s − 0.447·20-s + 1/5·25-s + 0.718·31-s − 1.41·32-s + 2/7·49-s + 0.282·50-s + 2.74·53-s + 1.01·62-s − 64-s − 0.450·79-s + 0.447·80-s + 0.404·98-s + 1/5·100-s + 3.88·106-s + 3.86·107-s − 1.63·121-s + 0.718·124-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.378634190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.378634190\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.406922182273769135817666765212, −7.74248215341661226066018130794, −7.36222834249445312924451264528, −6.91733127312419881675017505734, −6.42016330211366682483824205727, −6.04886632893443922618432703837, −5.38978411658686244811667052308, −5.23108444749782387813421742700, −4.43520770570742056751423466784, −4.23502207743543630151757117281, −3.65754163959813371874542409570, −3.12159764027016813824407272811, −2.57668628274294374485844424122, −1.92857169699302390151570766810, −0.72935820456086100637662106913,
0.72935820456086100637662106913, 1.92857169699302390151570766810, 2.57668628274294374485844424122, 3.12159764027016813824407272811, 3.65754163959813371874542409570, 4.23502207743543630151757117281, 4.43520770570742056751423466784, 5.23108444749782387813421742700, 5.38978411658686244811667052308, 6.04886632893443922618432703837, 6.42016330211366682483824205727, 6.91733127312419881675017505734, 7.36222834249445312924451264528, 7.74248215341661226066018130794, 8.406922182273769135817666765212
