| L(s) = 1 | + 0.510i·2-s + 1.73·4-s − 4.01·5-s + 1.90i·8-s − 2.04i·10-s − 4.55i·11-s + 1.36i·13-s + 2.50·16-s + 2.15·17-s + 2.66i·19-s − 6.99·20-s + 2.32·22-s + 1.27i·23-s + 11.1·25-s − 0.697·26-s + ⋯ |
| L(s) = 1 | + 0.360i·2-s + 0.869·4-s − 1.79·5-s + 0.674i·8-s − 0.647i·10-s − 1.37i·11-s + 0.379i·13-s + 0.626·16-s + 0.523·17-s + 0.610i·19-s − 1.56·20-s + 0.495·22-s + 0.266i·23-s + 2.22·25-s − 0.136·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.440023891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440023891\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.510iT - 2T^{2} \) |
| 5 | \( 1 + 4.01T + 5T^{2} \) |
| 11 | \( 1 + 4.55iT - 11T^{2} \) |
| 13 | \( 1 - 1.36iT - 13T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 - 2.66iT - 19T^{2} \) |
| 23 | \( 1 - 1.27iT - 23T^{2} \) |
| 29 | \( 1 + 8.75iT - 29T^{2} \) |
| 31 | \( 1 + 8.72iT - 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 + 7.31T + 43T^{2} \) |
| 47 | \( 1 - 8.97T + 47T^{2} \) |
| 53 | \( 1 - 7.89iT - 53T^{2} \) |
| 59 | \( 1 - 7.72T + 59T^{2} \) |
| 61 | \( 1 + 6.02iT - 61T^{2} \) |
| 67 | \( 1 - 8.15T + 67T^{2} \) |
| 71 | \( 1 - 0.301iT - 71T^{2} \) |
| 73 | \( 1 + 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 4.86T + 79T^{2} \) |
| 83 | \( 1 - 4.38T + 83T^{2} \) |
| 89 | \( 1 + 1.90T + 89T^{2} \) |
| 97 | \( 1 + 0.231iT - 97T^{2} \) |
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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
−9.532840902296761901850416839315, −8.364971734918948068968919963728, −7.898518860383336819117725258656, −7.42026374246993055677476252826, −6.30179167401492424846218926383, −5.68786036848640187559705079731, −4.27782124190475311772267765403, −3.58222757862647291141628168972, −2.57203689905081085489334758728, −0.73942466503382864274426343517,
1.06814618881972332785402494901, 2.59337351379022732638607645928, 3.47411331201527591723434237291, 4.33574795171249469550267993284, 5.27710462415728099340508375295, 6.82396882520063771975878071287, 7.14627844363065030198042758713, 7.87818725421239340699699489998, 8.715430304483984128577743444781, 9.851625359156830308954851348498
