| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 4·7-s + 8-s + 2·9-s − 3·11-s + 2·12-s − 3·13-s − 4·14-s − 16-s + 4·17-s − 2·18-s + 7·19-s − 8·21-s + 3·22-s + 3·23-s − 2·24-s − 4·25-s + 3·26-s − 6·27-s − 4·28-s + 15·29-s + 3·31-s + 5·32-s + 6·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 1.51·7-s + 0.353·8-s + 2/3·9-s − 0.904·11-s + 0.577·12-s − 0.832·13-s − 1.06·14-s − 1/4·16-s + 0.970·17-s − 0.471·18-s + 1.60·19-s − 1.74·21-s + 0.639·22-s + 0.625·23-s − 0.408·24-s − 4/5·25-s + 0.588·26-s − 1.15·27-s − 0.755·28-s + 2.78·29-s + 0.538·31-s + 0.883·32-s + 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354773 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354773 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 8 T + p T^{2} ) \) |
| 509 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 88 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T - 7 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 51 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 40 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 19 T + 198 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11 T + 189 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 148 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 19 T + 268 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
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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
−13.2322810690, −12.3673961199, −12.2198763176, −11.7555082514, −11.6043226881, −11.1587611124, −10.5185230730, −10.1707177561, −9.93250180883, −9.57267241936, −8.82964824231, −8.47523485872, −7.90167120991, −7.84271481164, −7.23017379110, −6.64267121436, −6.14903610365, −5.29777786901, −5.11305616255, −4.93869662946, −4.38063272344, −3.36114782322, −2.81236622297, −1.73442591146, −1.06663753443, 0,
1.06663753443, 1.73442591146, 2.81236622297, 3.36114782322, 4.38063272344, 4.93869662946, 5.11305616255, 5.29777786901, 6.14903610365, 6.64267121436, 7.23017379110, 7.84271481164, 7.90167120991, 8.47523485872, 8.82964824231, 9.57267241936, 9.93250180883, 10.1707177561, 10.5185230730, 11.1587611124, 11.6043226881, 11.7555082514, 12.2198763176, 12.3673961199, 13.2322810690
