| L(s) = 1 | − 2·2-s + 2·3-s + 2·4-s − 6·5-s − 4·6-s + 9-s + 12·10-s + 4·12-s − 12·15-s − 4·16-s + 4·17-s − 2·18-s − 12·20-s + 17·25-s + 2·27-s + 24·30-s + 2·31-s + 8·32-s − 8·34-s + 2·36-s − 6·45-s − 8·48-s + 2·49-s − 34·50-s + 8·51-s − 4·54-s − 10·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 4-s − 2.68·5-s − 1.63·6-s + 1/3·9-s + 3.79·10-s + 1.15·12-s − 3.09·15-s − 16-s + 0.970·17-s − 0.471·18-s − 2.68·20-s + 17/5·25-s + 0.384·27-s + 4.38·30-s + 0.359·31-s + 1.41·32-s − 1.37·34-s + 1/3·36-s − 0.894·45-s − 1.15·48-s + 2/7·49-s − 4.80·50-s + 1.12·51-s − 0.544·54-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$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 125 T^{2} + p^{2} T^{4} \) |
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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
−8.031225273898781861735996180295, −7.79563066894761764699076744722, −7.63972363858411562180892335667, −7.12213879792449961032253834807, −6.75115984477836891459169711463, −6.01322688301366249441623974499, −5.23257540219877286706400744895, −4.47515800107583670892532625284, −4.25972133216252406195846589916, −3.69681531446248613586154259491, −3.06355893347126638030817661746, −2.86475724713691549975182215516, −1.78207003119429557239715315798, −0.887583161107337569026694041378, 0,
0.887583161107337569026694041378, 1.78207003119429557239715315798, 2.86475724713691549975182215516, 3.06355893347126638030817661746, 3.69681531446248613586154259491, 4.25972133216252406195846589916, 4.47515800107583670892532625284, 5.23257540219877286706400744895, 6.01322688301366249441623974499, 6.75115984477836891459169711463, 7.12213879792449961032253834807, 7.63972363858411562180892335667, 7.79563066894761764699076744722, 8.031225273898781861735996180295
