| L(s) = 1 | + 2·3-s + 9-s − 6·11-s − 6·17-s + 8·19-s + 2·25-s − 4·27-s − 12·33-s + 8·41-s − 4·43-s + 10·49-s − 12·51-s + 16·57-s + 4·59-s − 4·67-s − 8·73-s + 4·75-s − 11·81-s − 12·83-s − 16·89-s + 12·97-s − 6·99-s + 2·107-s − 24·113-s + 6·121-s + 16·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.80·11-s − 1.45·17-s + 1.83·19-s + 2/5·25-s − 0.769·27-s − 2.08·33-s + 1.24·41-s − 0.609·43-s + 10/7·49-s − 1.68·51-s + 2.11·57-s + 0.520·59-s − 0.488·67-s − 0.936·73-s + 0.461·75-s − 1.22·81-s − 1.31·83-s − 1.69·89-s + 1.21·97-s − 0.603·99-s + 0.193·107-s − 2.25·113-s + 6/11·121-s + 1.44·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.76807781068152551394867832416, −7.50645284653553447969918055407, −7.01851001606684029524276604378, −6.64653494997348094411981500130, −5.81499474352357242880770811310, −5.51118217890980730414791550895, −5.20175235312312595489263966758, −4.41022256081360572581950518140, −4.21463886194424746897536034614, −3.33720913894854359266725310530, −3.00220192622829336791652233394, −2.51906657609184682950151358929, −2.15668454475295264960857664415, −1.17822521772056448827193503262, 0,
1.17822521772056448827193503262, 2.15668454475295264960857664415, 2.51906657609184682950151358929, 3.00220192622829336791652233394, 3.33720913894854359266725310530, 4.21463886194424746897536034614, 4.41022256081360572581950518140, 5.20175235312312595489263966758, 5.51118217890980730414791550895, 5.81499474352357242880770811310, 6.64653494997348094411981500130, 7.01851001606684029524276604378, 7.50645284653553447969918055407, 7.76807781068152551394867832416
