| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 5·7-s + 8-s + 9-s − 4·11-s − 2·12-s + 6·13-s + 5·14-s + 16-s − 6·17-s + 18-s − 2·19-s − 10·21-s − 4·22-s + 5·23-s − 2·24-s + 25-s + 6·26-s − 2·27-s + 5·28-s + 14·31-s + 32-s + 8·33-s − 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 1.88·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 1.66·13-s + 1.33·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.458·19-s − 2.18·21-s − 0.852·22-s + 1.04·23-s − 0.408·24-s + 1/5·25-s + 1.17·26-s − 0.384·27-s + 0.944·28-s + 2.51·31-s + 0.176·32-s + 1.39·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.844927420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.844927420\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 163 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T - 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−14.1065845275, −13.6927588279, −13.4314607769, −13.1781538817, −12.3664349478, −11.9989719764, −11.5019388662, −11.1764922411, −10.9377232049, −10.5255132440, −10.1688575662, −9.06757723706, −8.57249933375, −8.24716026120, −7.80313132311, −6.93045676289, −6.63214161334, −5.92186832869, −5.45247701395, −5.05904161934, −4.44797997287, −4.10410366168, −2.90253143149, −2.13286335703, −1.10137316308,
1.10137316308, 2.13286335703, 2.90253143149, 4.10410366168, 4.44797997287, 5.05904161934, 5.45247701395, 5.92186832869, 6.63214161334, 6.93045676289, 7.80313132311, 8.24716026120, 8.57249933375, 9.06757723706, 10.1688575662, 10.5255132440, 10.9377232049, 11.1764922411, 11.5019388662, 11.9989719764, 12.3664349478, 13.1781538817, 13.4314607769, 13.6927588279, 14.1065845275
