L(s) = 1 | + 3·3-s + 4-s + 2·7-s + 6·9-s + 3·12-s − 2·13-s + 16-s + 4·19-s + 6·21-s − 10·25-s + 9·27-s + 2·28-s + 2·31-s + 6·36-s + 14·37-s − 6·39-s − 24·43-s + 3·48-s + 3·49-s − 2·52-s + 12·57-s + 26·61-s + 12·63-s + 64-s + 22·67-s + 14·73-s − 30·75-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 1/2·4-s + 0.755·7-s + 2·9-s + 0.866·12-s − 0.554·13-s + 1/4·16-s + 0.917·19-s + 1.30·21-s − 2·25-s + 1.73·27-s + 0.377·28-s + 0.359·31-s + 36-s + 2.30·37-s − 0.960·39-s − 3.65·43-s + 0.433·48-s + 3/7·49-s − 0.277·52-s + 1.58·57-s + 3.32·61-s + 1.51·63-s + 1/8·64-s + 2.68·67-s + 1.63·73-s − 3.46·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.309745717\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.309745717\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−8.469152938024698877370110317561, −8.410332621243878174631771496694, −7.959201586617154759738546904094, −7.68409626813723475327909628947, −6.90643848137944117202320139495, −6.90516587374622685438056526369, −5.98244664513634320441588551536, −5.36737041037488579734906212522, −4.88738837750963504497893956492, −4.15982302022081941279948742151, −3.71179494687009087455073957292, −3.16464702872454555446785806554, −2.36943591207239051047957475876, −2.10681498002416439479097053291, −1.23766611312278004155286222469,
1.23766611312278004155286222469, 2.10681498002416439479097053291, 2.36943591207239051047957475876, 3.16464702872454555446785806554, 3.71179494687009087455073957292, 4.15982302022081941279948742151, 4.88738837750963504497893956492, 5.36737041037488579734906212522, 5.98244664513634320441588551536, 6.90516587374622685438056526369, 6.90643848137944117202320139495, 7.68409626813723475327909628947, 7.959201586617154759738546904094, 8.410332621243878174631771496694, 8.469152938024698877370110317561