| L(s) = 1 | + 2·3-s − 4·7-s + 9-s + 12·13-s − 16·19-s − 8·21-s + 25-s − 4·27-s + 8·31-s − 4·37-s + 24·39-s + 4·43-s − 2·49-s − 32·57-s − 4·61-s − 4·63-s + 12·67-s + 20·73-s + 2·75-s − 16·79-s − 11·81-s − 48·91-s + 16·93-s + 20·97-s + 4·103-s + 28·109-s − 8·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s + 1/3·9-s + 3.32·13-s − 3.67·19-s − 1.74·21-s + 1/5·25-s − 0.769·27-s + 1.43·31-s − 0.657·37-s + 3.84·39-s + 0.609·43-s − 2/7·49-s − 4.23·57-s − 0.512·61-s − 0.503·63-s + 1.46·67-s + 2.34·73-s + 0.230·75-s − 1.80·79-s − 1.22·81-s − 5.03·91-s + 1.65·93-s + 2.03·97-s + 0.394·103-s + 2.68·109-s − 0.759·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.369748295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.369748295\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.239480380770897255724570167213, −8.198425204155087916449678449322, −7.22523714288498578586167895314, −6.60040674980892955704654655808, −6.43994745928577778528949953628, −6.06254412327818625828699807195, −5.87904549724101637062073659041, −4.79631332498560161768109396865, −4.09416326053364009314265749502, −3.97804609056765892400553483994, −3.28314496469795666027926296042, −3.15160418737570888427061443506, −2.21397268061529340725377684972, −1.81188448910292297689023276847, −0.66259622561650473080382665441,
0.66259622561650473080382665441, 1.81188448910292297689023276847, 2.21397268061529340725377684972, 3.15160418737570888427061443506, 3.28314496469795666027926296042, 3.97804609056765892400553483994, 4.09416326053364009314265749502, 4.79631332498560161768109396865, 5.87904549724101637062073659041, 6.06254412327818625828699807195, 6.43994745928577778528949953628, 6.60040674980892955704654655808, 7.22523714288498578586167895314, 8.198425204155087916449678449322, 8.239480380770897255724570167213
