L(s) = 1 | − 2-s − 4·3-s + 4-s + 4·6-s − 8-s + 6·9-s − 4·12-s − 8·13-s + 16-s − 6·18-s + 4·24-s − 5·25-s + 8·26-s + 4·27-s − 8·31-s − 32-s + 6·36-s + 4·37-s + 32·39-s + 12·41-s + 16·43-s − 4·48-s + 49-s + 5·50-s − 8·52-s + 12·53-s − 4·54-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s + 1.63·6-s − 0.353·8-s + 2·9-s − 1.15·12-s − 2.21·13-s + 1/4·16-s − 1.41·18-s + 0.816·24-s − 25-s + 1.56·26-s + 0.769·27-s − 1.43·31-s − 0.176·32-s + 36-s + 0.657·37-s + 5.12·39-s + 1.87·41-s + 2.43·43-s − 0.577·48-s + 1/7·49-s + 0.707·50-s − 1.10·52-s + 1.64·53-s − 0.544·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−9.064899952938121589231451273163, −8.736817921425390383326382014904, −7.64673749340277573559389083829, −7.57571100088867902110310233811, −7.11642649742737663498250406032, −6.44599245093358620926386751475, −5.94966032900014960440899159541, −5.57928681742950427486583645839, −5.25236196288918634737499069062, −4.53924449636998865412903449029, −4.05250708127215343265869185182, −2.75791747804536479555350080645, −2.25326061715554950974013345200, −0.856225609282350254327766725168, 0,
0.856225609282350254327766725168, 2.25326061715554950974013345200, 2.75791747804536479555350080645, 4.05250708127215343265869185182, 4.53924449636998865412903449029, 5.25236196288918634737499069062, 5.57928681742950427486583645839, 5.94966032900014960440899159541, 6.44599245093358620926386751475, 7.11642649742737663498250406032, 7.57571100088867902110310233811, 7.64673749340277573559389083829, 8.736817921425390383326382014904, 9.064899952938121589231451273163