| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 2·5-s + 4·6-s + 2·7-s + 2·9-s + 4·10-s + 2·11-s − 4·12-s − 6·13-s − 4·14-s + 4·15-s − 4·16-s − 4·18-s − 6·19-s − 4·20-s − 4·21-s − 4·22-s − 25-s + 12·26-s − 6·27-s + 4·28-s − 2·29-s − 8·30-s − 16·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 0.894·5-s + 1.63·6-s + 0.755·7-s + 2/3·9-s + 1.26·10-s + 0.603·11-s − 1.15·12-s − 1.66·13-s − 1.06·14-s + 1.03·15-s − 16-s − 0.942·18-s − 1.37·19-s − 0.894·20-s − 0.872·21-s − 0.852·22-s − 1/5·25-s + 2.35·26-s − 1.15·27-s + 0.755·28-s − 0.371·29-s − 1.46·30-s − 2.87·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.46577793246840576941950435195, −10.36198207910143868452656178943, −9.491997576291117631391731084043, −9.367388518723168305184030040380, −8.773006473738189680563539278802, −8.398067639627249830401592268085, −7.82221982634645228963644952572, −7.40981490569013751150811715410, −6.97608401717528197695588646189, −6.96559094590727659148030102404, −5.73326828500251985233243200145, −5.71843200308178030147583344025, −4.92173408827166798320902009903, −4.24593149366840319265672024091, −4.16861259605725274387258399077, −3.09822321493158430231107950252, −1.84831417776244272968014517288, −1.75151869700390277834938640222, 0, 0,
1.75151869700390277834938640222, 1.84831417776244272968014517288, 3.09822321493158430231107950252, 4.16861259605725274387258399077, 4.24593149366840319265672024091, 4.92173408827166798320902009903, 5.71843200308178030147583344025, 5.73326828500251985233243200145, 6.96559094590727659148030102404, 6.97608401717528197695588646189, 7.40981490569013751150811715410, 7.82221982634645228963644952572, 8.398067639627249830401592268085, 8.773006473738189680563539278802, 9.367388518723168305184030040380, 9.491997576291117631391731084043, 10.36198207910143868452656178943, 10.46577793246840576941950435195
