| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 6·9-s − 10-s + 12·13-s + 16-s + 6·18-s + 20-s + 25-s − 12·26-s + 16·31-s − 32-s − 6·36-s + 20·37-s − 40-s + 4·41-s − 8·43-s − 6·45-s + 49-s − 50-s + 12·52-s + 4·53-s − 16·62-s + 64-s + 12·65-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 2·9-s − 0.316·10-s + 3.32·13-s + 1/4·16-s + 1.41·18-s + 0.223·20-s + 1/5·25-s − 2.35·26-s + 2.87·31-s − 0.176·32-s − 36-s + 3.28·37-s − 0.158·40-s + 0.624·41-s − 1.21·43-s − 0.894·45-s + 1/7·49-s − 0.141·50-s + 1.66·52-s + 0.549·53-s − 2.03·62-s + 1/8·64-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.762170815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.762170815\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.342925504546681427970604924895, −8.089729535357517933440155073370, −7.63810043151096799752774274091, −6.56718920518778472538358027386, −6.50661147582867110066747443111, −6.01430128602422585899125610915, −5.82344169953684002039972565231, −5.35297783463712406492518278139, −4.36823223624486251008730105784, −4.07505442753619333270812979660, −3.11732069850814337090071774581, −2.98711940319117381089884417433, −2.34423344984301921891298897662, −1.27008084899723889248060652436, −0.837608542604674414323310984226,
0.837608542604674414323310984226, 1.27008084899723889248060652436, 2.34423344984301921891298897662, 2.98711940319117381089884417433, 3.11732069850814337090071774581, 4.07505442753619333270812979660, 4.36823223624486251008730105784, 5.35297783463712406492518278139, 5.82344169953684002039972565231, 6.01430128602422585899125610915, 6.50661147582867110066747443111, 6.56718920518778472538358027386, 7.63810043151096799752774274091, 8.089729535357517933440155073370, 8.342925504546681427970604924895
