L(s) = 1 | − 2·2-s + 4-s − 3·5-s − 3·7-s − 4·9-s + 6·10-s + 11-s + 2·13-s + 6·14-s + 16-s + 2·17-s + 8·18-s − 4·19-s − 3·20-s − 2·22-s − 4·23-s − 2·25-s − 4·26-s − 3·28-s − 4·29-s − 3·31-s + 2·32-s − 4·34-s + 9·35-s − 4·36-s − 5·37-s + 8·38-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.34·5-s − 1.13·7-s − 4/3·9-s + 1.89·10-s + 0.301·11-s + 0.554·13-s + 1.60·14-s + 1/4·16-s + 0.485·17-s + 1.88·18-s − 0.917·19-s − 0.670·20-s − 0.426·22-s − 0.834·23-s − 2/5·25-s − 0.784·26-s − 0.566·28-s − 0.742·29-s − 0.538·31-s + 0.353·32-s − 0.685·34-s + 1.52·35-s − 2/3·36-s − 0.821·37-s + 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101039 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101039 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 23 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 191 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) |
good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 4 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 30 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 40 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 188 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 112 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 100 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 94 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 134 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 186 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−14.3970331394, −14.2192551902, −13.6216001155, −13.0730707204, −12.6473583272, −12.0621505353, −11.7972212913, −11.3820591755, −10.9086274510, −10.3743168646, −9.85567846804, −9.53931318047, −8.83901336309, −8.71065121770, −8.28688285242, −7.66702145514, −7.48186053696, −6.52656075421, −6.12971023493, −5.74501560159, −4.76756810742, −3.99416300347, −3.42493264509, −3.05859256732, −1.76260484747, 0, 0,
1.76260484747, 3.05859256732, 3.42493264509, 3.99416300347, 4.76756810742, 5.74501560159, 6.12971023493, 6.52656075421, 7.48186053696, 7.66702145514, 8.28688285242, 8.71065121770, 8.83901336309, 9.53931318047, 9.85567846804, 10.3743168646, 10.9086274510, 11.3820591755, 11.7972212913, 12.0621505353, 12.6473583272, 13.0730707204, 13.6216001155, 14.2192551902, 14.3970331394