| L(s) = 1 | + 2-s − 6·3-s + 4-s − 5-s − 6·6-s + 8-s + 21·9-s − 10-s − 6·12-s − 2·13-s + 6·15-s + 16-s + 21·18-s − 20-s − 6·24-s − 4·25-s − 2·26-s − 54·27-s + 6·30-s + 8·31-s + 32-s + 21·36-s + 6·37-s + 12·39-s − 40-s − 10·43-s − 21·45-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 3.46·3-s + 1/2·4-s − 0.447·5-s − 2.44·6-s + 0.353·8-s + 7·9-s − 0.316·10-s − 1.73·12-s − 0.554·13-s + 1.54·15-s + 1/4·16-s + 4.94·18-s − 0.223·20-s − 1.22·24-s − 4/5·25-s − 0.392·26-s − 10.3·27-s + 1.09·30-s + 1.43·31-s + 0.176·32-s + 7/2·36-s + 0.986·37-s + 1.92·39-s − 0.158·40-s − 1.52·43-s − 3.13·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.947525996725592169519910140162, −7.45985560667588965978165906144, −7.06351514889423061352505396698, −6.57298406981565936099456689488, −6.32528200643262897349180267262, −5.88908450915949561478136129218, −5.41696520049424981149332978983, −5.09892068910016055764395140389, −4.61153453491182141362175201622, −4.28784246485114607808557377793, −3.74839125626037417787439212589, −2.72660213623419522592921634901, −1.67224901915905998978362533056, −0.881323602688431658600765351508, 0,
0.881323602688431658600765351508, 1.67224901915905998978362533056, 2.72660213623419522592921634901, 3.74839125626037417787439212589, 4.28784246485114607808557377793, 4.61153453491182141362175201622, 5.09892068910016055764395140389, 5.41696520049424981149332978983, 5.88908450915949561478136129218, 6.32528200643262897349180267262, 6.57298406981565936099456689488, 7.06351514889423061352505396698, 7.45985560667588965978165906144, 7.947525996725592169519910140162
