| L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s + 3·10-s + 16-s + 5·19-s − 3·20-s + 3·23-s + 4·25-s − 9·29-s − 32-s − 5·38-s + 3·40-s − 9·43-s − 3·46-s − 15·47-s + 4·49-s − 4·50-s + 18·53-s + 9·58-s + 64-s + 18·67-s − 9·71-s − 27·73-s + 5·76-s − 3·80-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 1/4·16-s + 1.14·19-s − 0.670·20-s + 0.625·23-s + 4/5·25-s − 1.67·29-s − 0.176·32-s − 0.811·38-s + 0.474·40-s − 1.37·43-s − 0.442·46-s − 2.18·47-s + 4/7·49-s − 0.565·50-s + 2.47·53-s + 1.18·58-s + 1/8·64-s + 2.19·67-s − 1.06·71-s − 3.16·73-s + 0.573·76-s − 0.335·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.697195746157920782875702812837, −8.255504638981699164869591310184, −7.70968573373049697344916592480, −7.34107847737531780917400384171, −7.07166907084479305676430039793, −6.51718624391523389418725774430, −5.71974941571566655766109684626, −5.34729709925935676627870313075, −4.69327716580066576617467360574, −4.02815771306921680916527359686, −3.44958169279328725318537128187, −3.07375791172403503107853326193, −2.09245008436728335076150952409, −1.15529697078920214006424913776, 0,
1.15529697078920214006424913776, 2.09245008436728335076150952409, 3.07375791172403503107853326193, 3.44958169279328725318537128187, 4.02815771306921680916527359686, 4.69327716580066576617467360574, 5.34729709925935676627870313075, 5.71974941571566655766109684626, 6.51718624391523389418725774430, 7.07166907084479305676430039793, 7.34107847737531780917400384171, 7.70968573373049697344916592480, 8.255504638981699164869591310184, 8.697195746157920782875702812837
