| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 3·8-s + 9-s − 10-s − 12-s − 4·13-s − 15-s − 16-s − 4·17-s + 18-s + 8·19-s + 20-s − 3·24-s + 25-s − 4·26-s + 27-s + 4·29-s − 30-s + 5·32-s − 4·34-s − 36-s − 20·37-s + 8·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 1.10·13-s − 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.612·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.742·29-s − 0.182·30-s + 0.883·32-s − 0.685·34-s − 1/6·36-s − 3.28·37-s + 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.603932276417102120989834189554, −8.554588868470699894187777077269, −7.84465462281701093084420479537, −7.39546104131459972836674622338, −6.76955885158912340897756950945, −6.60942929553535216154216963636, −5.59929057414059862389948014289, −5.04860090093668311608273372623, −4.97287019204270070424993179755, −4.17881549969972779526027182436, −3.63412818236731287070311550143, −3.10526976657196724130917180316, −2.58465419848810939203642390782, −1.51207711394055808445468187309, 0,
1.51207711394055808445468187309, 2.58465419848810939203642390782, 3.10526976657196724130917180316, 3.63412818236731287070311550143, 4.17881549969972779526027182436, 4.97287019204270070424993179755, 5.04860090093668311608273372623, 5.59929057414059862389948014289, 6.60942929553535216154216963636, 6.76955885158912340897756950945, 7.39546104131459972836674622338, 7.84465462281701093084420479537, 8.554588868470699894187777077269, 8.603932276417102120989834189554
