| L(s) = 1 | − 2-s − 2·3-s + 2·6-s − 2·7-s − 8-s + 3·9-s − 6·11-s − 2·13-s + 2·14-s − 16-s − 2·17-s − 3·18-s + 6·19-s + 4·21-s + 6·22-s − 2·23-s + 2·24-s + 2·26-s − 4·27-s − 8·29-s + 6·32-s + 12·33-s + 2·34-s − 6·37-s − 6·38-s + 4·39-s − 4·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 0.816·6-s − 0.755·7-s − 0.353·8-s + 9-s − 1.80·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.707·18-s + 1.37·19-s + 0.872·21-s + 1.27·22-s − 0.417·23-s + 0.408·24-s + 0.392·26-s − 0.769·27-s − 1.48·29-s + 1.06·32-s + 2.08·33-s + 0.342·34-s − 0.986·37-s − 0.973·38-s + 0.640·39-s − 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 24 T + 265 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 206 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−10.35569779149786391198674832562, −10.19076216564603921363296522408, −9.951698078233311084736302459695, −9.449000515301773254934513204734, −8.866581983686754893081788832593, −8.597131183005512731294545054797, −7.71818053244245840677303402626, −7.55204128151397839282240414229, −7.10402450061182345710764698096, −6.44344883082224725355775889897, −6.07322163374761089930402762178, −5.51108691449951128018911820373, −4.92286902214624552193729810950, −4.85427238984954742942375904518, −3.76119769020099550213146009135, −3.16185744959590315349643963998, −2.50889448580393586220484289268, −1.58434951430001554434261075014, 0, 0,
1.58434951430001554434261075014, 2.50889448580393586220484289268, 3.16185744959590315349643963998, 3.76119769020099550213146009135, 4.85427238984954742942375904518, 4.92286902214624552193729810950, 5.51108691449951128018911820373, 6.07322163374761089930402762178, 6.44344883082224725355775889897, 7.10402450061182345710764698096, 7.55204128151397839282240414229, 7.71818053244245840677303402626, 8.597131183005512731294545054797, 8.866581983686754893081788832593, 9.449000515301773254934513204734, 9.951698078233311084736302459695, 10.19076216564603921363296522408, 10.35569779149786391198674832562
