| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s − 4·5-s + 8·6-s − 6·7-s + 6·9-s + 8·10-s − 2·11-s − 8·12-s + 12·14-s + 16·15-s − 4·16-s + 2·17-s − 12·18-s − 6·19-s − 8·20-s + 24·21-s + 4·22-s − 2·23-s + 11·25-s + 4·27-s − 12·28-s − 14·29-s − 32·30-s + 8·32-s + 8·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s − 1.78·5-s + 3.26·6-s − 2.26·7-s + 2·9-s + 2.52·10-s − 0.603·11-s − 2.30·12-s + 3.20·14-s + 4.13·15-s − 16-s + 0.485·17-s − 2.82·18-s − 1.37·19-s − 1.78·20-s + 5.23·21-s + 0.852·22-s − 0.417·23-s + 11/5·25-s + 0.769·27-s − 2.26·28-s − 2.59·29-s − 5.84·30-s + 1.41·32-s + 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−14.21442938078161757019809759424, −12.96317739991569446309415491808, −12.80830581494530963976800378598, −12.43225073815099575731022978335, −11.56026651288003495969755770482, −11.43204301764673511085392549231, −10.68161814271764683129864823345, −10.61649948138300935650475551650, −9.699859526262278865980790027689, −9.307682822426627168383887243707, −8.294765114665058736602328411165, −7.88638268394394799902523829232, −6.88805760583972888980449229505, −6.69179242865704678912310909522, −5.99255617960231670738492661515, −5.18013618333723223342449259876, −4.16929381792922326649904005931, −3.19744395278082895586299779894, 0, 0,
3.19744395278082895586299779894, 4.16929381792922326649904005931, 5.18013618333723223342449259876, 5.99255617960231670738492661515, 6.69179242865704678912310909522, 6.88805760583972888980449229505, 7.88638268394394799902523829232, 8.294765114665058736602328411165, 9.307682822426627168383887243707, 9.699859526262278865980790027689, 10.61649948138300935650475551650, 10.68161814271764683129864823345, 11.43204301764673511085392549231, 11.56026651288003495969755770482, 12.43225073815099575731022978335, 12.80830581494530963976800378598, 12.96317739991569446309415491808, 14.21442938078161757019809759424
