L(s) = 1 | − 4.28·2-s − 2.85i·3-s + 10.3·4-s + 6.36i·5-s + 12.2i·6-s + 29.4i·7-s − 10.2·8-s + 18.8·9-s − 27.3i·10-s + 61.3i·11-s − 29.6i·12-s + 18.5·13-s − 126. i·14-s + 18.1·15-s − 39.1·16-s + ⋯ |
L(s) = 1 | − 1.51·2-s − 0.548i·3-s + 1.29·4-s + 0.569i·5-s + 0.831i·6-s + 1.58i·7-s − 0.453·8-s + 0.699·9-s − 0.863i·10-s + 1.68i·11-s − 0.712i·12-s + 0.395·13-s − 2.40i·14-s + 0.312·15-s − 0.611·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5875495455\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5875495455\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 + 4.28T + 8T^{2} \) |
| 3 | \( 1 + 2.85iT - 27T^{2} \) |
| 5 | \( 1 - 6.36iT - 125T^{2} \) |
| 7 | \( 1 - 29.4iT - 343T^{2} \) |
| 11 | \( 1 - 61.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 18.5T + 2.19e3T^{2} \) |
| 19 | \( 1 + 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.38iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 164. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 127. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 158. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 31.3iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 157.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 166.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 343.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 112. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 984.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 524. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 852. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 201. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 22.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 502.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 680. iT - 9.12e5T^{2} \) |
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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
−11.62959219296678053165285069646, −10.44178724256916062065940106565, −9.801033812479823998634505346345, −8.845088586584952638188431898710, −8.056120610585986127587364639683, −6.99872222267233978234692608597, −6.37829049555815908603153747535, −4.64233985225450683141899688347, −2.43291134722682744778379398346, −1.68083708856015830397038680867,
0.39765925977043359530612457843, 1.36014464833314687185882533811, 3.62080143464918254354128672601, 4.66598292163326709189945581641, 6.41248441336270205285311890597, 7.40226628067198032876990901463, 8.405351881331427309490982939408, 9.021518029633155539689343089514, 10.14577126258411283756696405826, 10.68981061326581586231440335587