| L(s) = 1 | + 3-s + 4-s + 4·7-s + 9-s + 12-s + 4·13-s + 16-s − 8·19-s + 4·21-s − 10·25-s + 27-s + 4·28-s − 20·31-s + 36-s + 16·37-s + 4·39-s − 8·43-s + 48-s − 2·49-s + 4·52-s − 8·57-s + 16·61-s + 4·63-s + 64-s − 8·67-s + 4·73-s − 10·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.51·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s − 1.83·19-s + 0.872·21-s − 2·25-s + 0.192·27-s + 0.755·28-s − 3.59·31-s + 1/6·36-s + 2.63·37-s + 0.640·39-s − 1.21·43-s + 0.144·48-s − 2/7·49-s + 0.554·52-s − 1.05·57-s + 2.04·61-s + 0.503·63-s + 1/8·64-s − 0.977·67-s + 0.468·73-s − 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31212 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31212 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.920811713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.920811713\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−10.57689980236595016687233880100, −10.00723343046726241119720950602, −9.361746587069330588975575945669, −8.745924693621362183100177299272, −8.367508603382134314146293072792, −7.81014283472581039979750102513, −7.50641789037179894281138679970, −6.70932306148654350305246926067, −5.96956680213085492365415051586, −5.60715979666329449780909401068, −4.60759445774375135200288872743, −4.05372696693836488551925909574, −3.42809860399723990451499242436, −2.04126125287275460099535053657, −1.81193945201237649695550582777,
1.81193945201237649695550582777, 2.04126125287275460099535053657, 3.42809860399723990451499242436, 4.05372696693836488551925909574, 4.60759445774375135200288872743, 5.60715979666329449780909401068, 5.96956680213085492365415051586, 6.70932306148654350305246926067, 7.50641789037179894281138679970, 7.81014283472581039979750102513, 8.367508603382134314146293072792, 8.745924693621362183100177299272, 9.361746587069330588975575945669, 10.00723343046726241119720950602, 10.57689980236595016687233880100
