| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 2·5-s + 4·6-s − 3·9-s − 4·10-s − 4·12-s − 4·15-s − 4·16-s − 4·17-s + 6·18-s + 4·20-s − 7·25-s + 14·27-s + 8·30-s + 14·31-s + 8·32-s + 8·34-s − 6·36-s − 6·45-s + 8·48-s − 10·49-s + 14·50-s + 8·51-s − 28·54-s + 10·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 0.894·5-s + 1.63·6-s − 9-s − 1.26·10-s − 1.15·12-s − 1.03·15-s − 16-s − 0.970·17-s + 1.41·18-s + 0.894·20-s − 7/5·25-s + 2.69·27-s + 1.46·30-s + 2.51·31-s + 1.41·32-s + 1.37·34-s − 36-s − 0.894·45-s + 1.15·48-s − 1.42·49-s + 1.97·50-s + 1.12·51-s − 3.81·54-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.449863588203762160392147957262, −7.71564009386520602031555979698, −7.26767839065965563915798103089, −6.55870218284239196343388663114, −6.36261389471308870138602900888, −6.05033687233339124910623524332, −5.45660481055156382794908917127, −5.04473970036247998468059015652, −4.52429902234882156392401341160, −3.86837030910736326609302269235, −2.79441437865048826367485081806, −2.49738876285848381191850982579, −1.74297906818372540900004970449, −0.836582978875406341548589152596, 0,
0.836582978875406341548589152596, 1.74297906818372540900004970449, 2.49738876285848381191850982579, 2.79441437865048826367485081806, 3.86837030910736326609302269235, 4.52429902234882156392401341160, 5.04473970036247998468059015652, 5.45660481055156382794908917127, 6.05033687233339124910623524332, 6.36261389471308870138602900888, 6.55870218284239196343388663114, 7.26767839065965563915798103089, 7.71564009386520602031555979698, 8.449863588203762160392147957262
