| L(s) = 1 | − 2-s − 2·3-s − 6·5-s + 2·6-s − 3·7-s + 8-s + 9-s + 6·10-s − 6·11-s − 4·13-s + 3·14-s + 12·15-s − 16-s − 18-s + 3·19-s + 6·21-s + 6·22-s − 3·23-s − 2·24-s + 17·25-s + 4·26-s + 4·27-s + 3·29-s − 12·30-s − 6·31-s + 12·33-s + 18·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 2.68·5-s + 0.816·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 1.89·10-s − 1.80·11-s − 1.10·13-s + 0.801·14-s + 3.09·15-s − 1/4·16-s − 0.235·18-s + 0.688·19-s + 1.30·21-s + 1.27·22-s − 0.625·23-s − 0.408·24-s + 17/5·25-s + 0.784·26-s + 0.769·27-s + 0.557·29-s − 2.19·30-s − 1.07·31-s + 2.08·33-s + 3.04·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18252 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18252 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T - 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T - 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 57 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 91 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 142 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−16.2764359745, −16.0205678159, −15.6425791200, −15.2719960102, −14.6556518095, −13.9807079849, −13.1531156694, −12.6591802500, −12.4011257037, −11.8307863750, −11.4967047581, −11.0309985917, −10.4355233020, −10.0333927408, −9.47917127190, −8.49070983347, −8.15060437286, −7.63114556384, −7.20344014291, −6.70057203308, −5.64195783829, −5.03718812702, −4.44467997201, −3.54343357851, −2.89217233037, 0, 0,
2.89217233037, 3.54343357851, 4.44467997201, 5.03718812702, 5.64195783829, 6.70057203308, 7.20344014291, 7.63114556384, 8.15060437286, 8.49070983347, 9.47917127190, 10.0333927408, 10.4355233020, 11.0309985917, 11.4967047581, 11.8307863750, 12.4011257037, 12.6591802500, 13.1531156694, 13.9807079849, 14.6556518095, 15.2719960102, 15.6425791200, 16.0205678159, 16.2764359745
