| L(s) = 1 | + 2-s − 2·3-s − 4-s − 2·6-s − 7-s − 3·8-s − 3·11-s + 2·12-s − 3·13-s − 14-s − 16-s − 3·17-s − 6·19-s + 2·21-s − 3·22-s − 3·23-s + 6·24-s − 4·25-s − 3·26-s + 2·27-s + 28-s + 7·29-s − 6·31-s + 5·32-s + 6·33-s − 3·34-s − 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s − 0.377·7-s − 1.06·8-s − 0.904·11-s + 0.577·12-s − 0.832·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s − 1.37·19-s + 0.436·21-s − 0.639·22-s − 0.625·23-s + 1.22·24-s − 4/5·25-s − 0.588·26-s + 0.384·27-s + 0.188·28-s + 1.29·29-s − 1.07·31-s + 0.883·32-s + 1.04·33-s − 0.514·34-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 251 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 15 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 24 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 79 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 19 T + 184 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 36 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 120 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T - 4 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11 T + 74 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 122 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−13.3554089479, −13.0829230462, −12.5437146329, −12.1546837006, −11.9583814995, −11.5493706329, −10.8812189126, −10.5940715728, −10.2948820124, −9.72501049184, −9.23390916581, −8.78306041286, −8.42530912114, −7.67412194253, −7.46300573131, −6.53998174244, −6.27081429264, −5.91386343038, −5.39907361605, −4.95617938097, −4.39433708636, −4.14503085804, −3.15106592446, −2.71783683441, −1.85867028299, 0, 0,
1.85867028299, 2.71783683441, 3.15106592446, 4.14503085804, 4.39433708636, 4.95617938097, 5.39907361605, 5.91386343038, 6.27081429264, 6.53998174244, 7.46300573131, 7.67412194253, 8.42530912114, 8.78306041286, 9.23390916581, 9.72501049184, 10.2948820124, 10.5940715728, 10.8812189126, 11.5493706329, 11.9583814995, 12.1546837006, 12.5437146329, 13.0829230462, 13.3554089479
