| L(s) = 1 | + 6.92·5-s + 12i·7-s − 6.92i·11-s − 13.8·13-s − 14·17-s − 34.6i·19-s + 24i·23-s + 22.9·25-s − 34.6·29-s + 12i·31-s + 83.1i·35-s − 27.7·37-s − 14·41-s + 6.92i·43-s + 72i·47-s + ⋯ |
| L(s) = 1 | + 1.38·5-s + 1.71i·7-s − 0.629i·11-s − 1.06·13-s − 0.823·17-s − 1.82i·19-s + 1.04i·23-s + 0.919·25-s − 1.19·29-s + 0.387i·31-s + 2.37i·35-s − 0.748·37-s − 0.341·41-s + 0.161i·43-s + 1.53i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5654288638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5654288638\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 6.92T + 25T^{2} \) |
| 7 | \( 1 - 12iT - 49T^{2} \) |
| 11 | \( 1 + 6.92iT - 121T^{2} \) |
| 13 | \( 1 + 13.8T + 169T^{2} \) |
| 17 | \( 1 + 14T + 289T^{2} \) |
| 19 | \( 1 + 34.6iT - 361T^{2} \) |
| 23 | \( 1 - 24iT - 529T^{2} \) |
| 29 | \( 1 + 34.6T + 841T^{2} \) |
| 31 | \( 1 - 12iT - 961T^{2} \) |
| 37 | \( 1 + 27.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.92iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 72iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 62.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 48.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 55.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 90.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 24iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 50T + 5.32e3T^{2} \) |
| 79 | \( 1 - 12iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 20.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 62T + 7.92e3T^{2} \) |
| 97 | \( 1 + 146T + 9.40e3T^{2} \) |
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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
−9.339040336747256734251968897144, −8.791075893572574366294498199819, −7.68655438974873554993153590168, −6.66524722603619548486813573592, −6.02172162805812740290092714039, −5.30402836287317077304614744900, −4.81446907129083433588258067280, −3.09619302499364497639361423965, −2.43850728468916313790295254397, −1.71213317403570871548063137943,
0.11631814328452418651769597117, 1.57256986404669006532199694205, 2.21141932682903369397591332452, 3.60770594327316661978364120785, 4.42907541494785007215318002647, 5.23722289348336981368932175379, 6.18506570500659644175479777943, 6.92564302222228067112601002645, 7.49634586852132764058862386280, 8.417515991598636750972250211050
