L(s) = 1 | + (−0.286 − 0.957i)2-s + (0.597 − 0.802i)3-s + (−0.835 + 0.549i)4-s + (0.396 + 0.918i)5-s + (−0.939 − 0.342i)6-s + (0.115 − 1.98i)7-s + (0.766 + 0.642i)8-s + (−0.286 − 0.957i)9-s + (0.766 − 0.642i)10-s + (−0.0581 + 0.998i)12-s + (−1.93 + 0.458i)14-s + (0.973 + 0.230i)15-s + (0.396 − 0.918i)16-s + (−0.835 + 0.549i)18-s + (−0.835 − 0.549i)20-s + (−1.52 − 1.27i)21-s + ⋯ |
L(s) = 1 | + (−0.286 − 0.957i)2-s + (0.597 − 0.802i)3-s + (−0.835 + 0.549i)4-s + (0.396 + 0.918i)5-s + (−0.939 − 0.342i)6-s + (0.115 − 1.98i)7-s + (0.766 + 0.642i)8-s + (−0.286 − 0.957i)9-s + (0.766 − 0.642i)10-s + (−0.0581 + 0.998i)12-s + (−1.93 + 0.458i)14-s + (0.973 + 0.230i)15-s + (0.396 − 0.918i)16-s + (−0.835 + 0.549i)18-s + (−0.835 − 0.549i)20-s + (−1.52 − 1.27i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.138371748\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.138371748\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.286 + 0.957i)T \) |
| 3 | \( 1 + (-0.597 + 0.802i)T \) |
| 5 | \( 1 + (-0.396 - 0.918i)T \) |
good | 7 | \( 1 + (-0.115 + 1.98i)T + (-0.993 - 0.116i)T^{2} \) |
| 11 | \( 1 + (0.286 - 0.957i)T^{2} \) |
| 13 | \( 1 + (0.0581 - 0.998i)T^{2} \) |
| 17 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 19 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.00676 - 0.116i)T + (-0.993 + 0.116i)T^{2} \) |
| 29 | \( 1 + (-1.73 - 0.412i)T + (0.893 + 0.448i)T^{2} \) |
| 31 | \( 1 + (-0.597 - 0.802i)T^{2} \) |
| 37 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 41 | \( 1 + (-0.393 + 1.31i)T + (-0.835 - 0.549i)T^{2} \) |
| 43 | \( 1 + (1.52 - 0.177i)T + (0.973 - 0.230i)T^{2} \) |
| 47 | \( 1 + (0.512 - 0.257i)T + (0.597 - 0.802i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.286 + 0.957i)T^{2} \) |
| 61 | \( 1 + (0.997 + 0.656i)T + (0.396 + 0.918i)T^{2} \) |
| 67 | \( 1 + (-1.89 + 0.448i)T + (0.893 - 0.448i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (0.835 - 0.549i)T^{2} \) |
| 83 | \( 1 + (-0.393 - 1.31i)T + (-0.835 + 0.549i)T^{2} \) |
| 89 | \( 1 + (0.439 + 0.368i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (0.686 + 0.727i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.531473626740412526151150631225, −8.419240879657696071503060967981, −7.73463226871897895727424021203, −7.05909655262213928757743131057, −6.43635800463329259624649523092, −4.85144713881267776670462000797, −3.74272829023577232788115549353, −3.20434694206702390984273578106, −2.05734214625405804406309294466, −0.991462626967129498985460659260,
1.84117353542762479266072955373, 2.99130120805570453415708010534, 4.49376011256734676647488967996, 5.02805541739001838853390845509, 5.73378536077026555573827902331, 6.46439462626130838772597362168, 7.972366883882439690244215199309, 8.493456796973468096205154491952, 8.854748386747692765417821633247, 9.676212252019975825349710551714