| L(s) = 1 | − 9-s + 4·13-s + 2·25-s − 49-s − 4·61-s + 81-s − 4·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 9-s + 4·13-s + 2·25-s − 49-s − 4·61-s + 81-s − 4·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.272912672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.272912672\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−10.11200839042751556043772798304, −9.354954356029106958045375089875, −9.102455760899311584670151755378, −8.741660049457947189601230705677, −8.402172298659248132592523459432, −8.292130196846028609239823856448, −7.67004482517405927059557654228, −7.13977745847292536294915545202, −6.42183063986854377059349482217, −6.38460940670539142674041580352, −5.82341410171504914367181872037, −5.77339543976341204223818388542, −4.79153249099958305568443874700, −4.65190651024797205426139835890, −3.80971667961244198549523359263, −3.32011807519595132741865495142, −3.28439258559214431625703184604, −2.47495016111105270558228759483, −1.40486403518532242249004867753, −1.20403815503451087888988435018,
1.20403815503451087888988435018, 1.40486403518532242249004867753, 2.47495016111105270558228759483, 3.28439258559214431625703184604, 3.32011807519595132741865495142, 3.80971667961244198549523359263, 4.65190651024797205426139835890, 4.79153249099958305568443874700, 5.77339543976341204223818388542, 5.82341410171504914367181872037, 6.38460940670539142674041580352, 6.42183063986854377059349482217, 7.13977745847292536294915545202, 7.67004482517405927059557654228, 8.292130196846028609239823856448, 8.402172298659248132592523459432, 8.741660049457947189601230705677, 9.102455760899311584670151755378, 9.354954356029106958045375089875, 10.11200839042751556043772798304
