| L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s − 2·9-s + 2·10-s − 2·13-s − 16-s + 4·17-s + 2·18-s + 2·20-s + 3·25-s + 2·26-s + 4·29-s − 5·32-s − 4·34-s + 2·36-s − 4·37-s − 6·40-s − 12·41-s + 4·45-s + 2·49-s − 3·50-s + 2·52-s + 4·53-s − 4·58-s + 4·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s − 2/3·9-s + 0.632·10-s − 0.554·13-s − 1/4·16-s + 0.970·17-s + 0.471·18-s + 0.447·20-s + 3/5·25-s + 0.392·26-s + 0.742·29-s − 0.883·32-s − 0.685·34-s + 1/3·36-s − 0.657·37-s − 0.948·40-s − 1.87·41-s + 0.596·45-s + 2/7·49-s − 0.424·50-s + 0.277·52-s + 0.549·53-s − 0.525·58-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.822167931588977599398615084740, −8.892263963780957263247682851131, −8.679150483937958571354498819427, −8.050663651453834316178513676641, −7.893612143808235569104846929392, −7.04231773996815600274218157258, −6.86847438549507433409027285363, −5.80677556175820100036715333304, −5.26806860730181407315208823154, −4.76761858373058400551305930705, −4.03550878858777485512286959394, −3.43626038544626492191293151701, −2.65504338827892484384496886087, −1.34491844974665976517057296331, 0,
1.34491844974665976517057296331, 2.65504338827892484384496886087, 3.43626038544626492191293151701, 4.03550878858777485512286959394, 4.76761858373058400551305930705, 5.26806860730181407315208823154, 5.80677556175820100036715333304, 6.86847438549507433409027285363, 7.04231773996815600274218157258, 7.893612143808235569104846929392, 8.050663651453834316178513676641, 8.679150483937958571354498819427, 8.892263963780957263247682851131, 9.822167931588977599398615084740
