| L(s) = 1 | − 2-s − 3-s − 4-s − 2·5-s + 6-s + 2·7-s + 3·8-s + 9-s + 2·10-s − 8·11-s + 12-s − 2·14-s + 2·15-s − 16-s − 12·17-s − 18-s + 2·20-s − 2·21-s + 8·22-s − 3·24-s − 25-s − 27-s − 2·28-s − 2·30-s − 5·32-s + 8·33-s + 12·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 2.41·11-s + 0.288·12-s − 0.534·14-s + 0.516·15-s − 1/4·16-s − 2.91·17-s − 0.235·18-s + 0.447·20-s − 0.436·21-s + 1.70·22-s − 0.612·24-s − 1/5·25-s − 0.192·27-s − 0.377·28-s − 0.365·30-s − 0.883·32-s + 1.39·33-s + 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−8.068098981215555715874277637796, −7.65494363024181651773191318169, −7.38325958034671995701748313853, −6.85360812870284917864517668882, −6.19643457342138342025418009798, −5.61533530390354367379245545667, −4.94701466488767178340177905762, −4.87983646890892800864940889103, −4.13981751462080886088964913424, −4.03796996879678948025961788837, −2.72775334283054769167882483900, −2.39234840548789475685588947322, −1.46036150187712502072371434143, 0, 0,
1.46036150187712502072371434143, 2.39234840548789475685588947322, 2.72775334283054769167882483900, 4.03796996879678948025961788837, 4.13981751462080886088964913424, 4.87983646890892800864940889103, 4.94701466488767178340177905762, 5.61533530390354367379245545667, 6.19643457342138342025418009798, 6.85360812870284917864517668882, 7.38325958034671995701748313853, 7.65494363024181651773191318169, 8.068098981215555715874277637796
