L(s) = 1 | − 0.414i·2-s + 1.82·4-s − 2.93·5-s − 1.58i·8-s + 1.21i·10-s − 4.82i·11-s − 2.93i·13-s + 3·16-s + 7.07·17-s − 5.86i·19-s − 5.35·20-s − 1.99·22-s + 2i·23-s + 3.58·25-s − 1.21·26-s + ⋯ |
L(s) = 1 | − 0.292i·2-s + 0.914·4-s − 1.31·5-s − 0.560i·8-s + 0.383i·10-s − 1.45i·11-s − 0.812i·13-s + 0.750·16-s + 1.71·17-s − 1.34i·19-s − 1.19·20-s − 0.426·22-s + 0.417i·23-s + 0.717·25-s − 0.238·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04151 - 0.856797i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04151 - 0.856797i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.414iT - 2T^{2} \) |
| 5 | \( 1 + 2.93T + 5T^{2} \) |
| 11 | \( 1 + 4.82iT - 11T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 + 5.86iT - 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 - 0.828iT - 29T^{2} \) |
| 31 | \( 1 - 5.86iT - 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 + 5.86T + 47T^{2} \) |
| 53 | \( 1 - 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 5.86T + 59T^{2} \) |
| 61 | \( 1 + 1.21iT - 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 + 0.828iT - 71T^{2} \) |
| 73 | \( 1 + 7.07iT - 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 7.07iT - 97T^{2} \) |
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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
−10.95015956061468846404655326536, −10.45591812932921763874951323714, −9.034598626197164437101200964727, −7.945504455042140796031363916372, −7.46760711251867759034489938905, −6.27045323738108342021502708602, −5.18846765162293514375990553507, −3.50969728613556461161087510119, −3.06899434770365071138075860206, −0.895127659542059370612270288503,
1.82239806321550030655164306323, 3.40483890303602421570490644077, 4.43129745911099922223459928746, 5.75289353857816803888460368893, 6.91755309503203448790411946531, 7.63116457889550279286929469454, 8.151973433332395515282422819242, 9.658260898346562637287685039977, 10.45058892088799411425187524169, 11.55633290914218105338643643009