| L(s) = 1 | − 7.37i·5-s − 1.26·7-s + 5.82i·11-s − 10.4·13-s − 18.3i·17-s + 20.8·19-s − 20.4i·23-s − 29.4·25-s − 11.1i·29-s − 61.3·31-s + 9.34i·35-s − 38.4·37-s + 33.0i·41-s + 49.3·43-s + 21.5i·47-s + ⋯ |
| L(s) = 1 | − 1.47i·5-s − 0.180·7-s + 0.529i·11-s − 0.806·13-s − 1.07i·17-s + 1.09·19-s − 0.890i·23-s − 1.17·25-s − 0.385i·29-s − 1.97·31-s + 0.266i·35-s − 1.03·37-s + 0.807i·41-s + 1.14·43-s + 0.459i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6976791944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6976791944\) |
| \(L(2)\) |
|
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 | \( 1 + 7.37iT - 25T^{2} \) |
| 7 | \( 1 + 1.26T + 49T^{2} \) |
| 11 | \( 1 - 5.82iT - 121T^{2} \) |
| 13 | \( 1 + 10.4T + 169T^{2} \) |
| 17 | \( 1 + 18.3iT - 289T^{2} \) |
| 19 | \( 1 - 20.8T + 361T^{2} \) |
| 23 | \( 1 + 20.4iT - 529T^{2} \) |
| 29 | \( 1 + 11.1iT - 841T^{2} \) |
| 31 | \( 1 + 61.3T + 961T^{2} \) |
| 37 | \( 1 + 38.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 21.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 77.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 25.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 55.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 91.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 114. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 8.21T + 6.24e3T^{2} \) |
| 83 | \( 1 - 150. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 72.8T + 9.40e3T^{2} \) |
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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
−9.285875334223890016512628575499, −9.028087359753621783142935207468, −7.75530734662898785879730821170, −7.20687828238217380201164172402, −5.82807603334513713259212073190, −4.97733505047026042622548641070, −4.38742043845337020421221463021, −2.92337120602865015632877475591, −1.53509448152557832215379165094, −0.22276547722026821297372932774,
1.85788278309711083591505916152, 3.12452759142426931361158168411, 3.71445026583041695714130360251, 5.30326877839669832938546775647, 6.08044778575860359229658166777, 7.15253887177938312842293795610, 7.50684375565797054917590244145, 8.764351562935074975767576608645, 9.697624923657406961641369923414, 10.45562802845487161997066390947
