| L(s) = 1 | − 2·3-s + 4·5-s + 3·9-s + 8·13-s − 8·15-s + 11·25-s − 4·27-s + 16·31-s − 8·37-s − 16·39-s − 12·41-s + 8·43-s + 12·45-s − 2·49-s + 24·53-s + 32·65-s + 24·67-s − 32·71-s − 22·75-s − 16·79-s + 5·81-s + 24·83-s − 20·89-s − 32·93-s − 24·107-s + 16·111-s + 24·117-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 9-s + 2.21·13-s − 2.06·15-s + 11/5·25-s − 0.769·27-s + 2.87·31-s − 1.31·37-s − 2.56·39-s − 1.87·41-s + 1.21·43-s + 1.78·45-s − 2/7·49-s + 3.29·53-s + 3.96·65-s + 2.93·67-s − 3.79·71-s − 2.54·75-s − 1.80·79-s + 5/9·81-s + 2.63·83-s − 2.11·89-s − 3.31·93-s − 2.32·107-s + 1.51·111-s + 2.21·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.637005526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.637005526\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.643341868131511253919080652131, −8.439280947041724828369778986336, −8.174774246779822126799057410626, −7.28467451916425409737504924828, −7.03867932419971488116179221426, −6.71883504358325432517094108402, −6.19659691040763031589308609790, −6.19449141239918099876392670374, −5.76603369735828493651123605100, −5.39294326186920822235473980826, −5.12563768601513364714137869726, −4.62351507821192938027965605164, −4.11529057434854098221539106148, −3.77837432412368337841722462532, −3.15912533813683896638130809239, −2.63868692538247275934287778627, −2.18095538367954224802715980440, −1.38670433189134003176200789978, −1.30946335159695553694491586273, −0.66490024439535186336475027711,
0.66490024439535186336475027711, 1.30946335159695553694491586273, 1.38670433189134003176200789978, 2.18095538367954224802715980440, 2.63868692538247275934287778627, 3.15912533813683896638130809239, 3.77837432412368337841722462532, 4.11529057434854098221539106148, 4.62351507821192938027965605164, 5.12563768601513364714137869726, 5.39294326186920822235473980826, 5.76603369735828493651123605100, 6.19449141239918099876392670374, 6.19659691040763031589308609790, 6.71883504358325432517094108402, 7.03867932419971488116179221426, 7.28467451916425409737504924828, 8.174774246779822126799057410626, 8.439280947041724828369778986336, 8.643341868131511253919080652131
