| L(s) = 1 | + (−0.781 − 1.17i)2-s + (2.14 + 2.61i)3-s + (−0.779 + 1.84i)4-s + (−1.87 − 0.568i)5-s + (1.40 − 4.57i)6-s + (1.31 − 0.260i)7-s + (2.78 − 0.519i)8-s + (−1.64 + 8.28i)9-s + (0.793 + 2.65i)10-s + (1.25 − 0.123i)11-s + (−6.48 + 1.91i)12-s + (0.385 + 1.27i)13-s + (−1.33 − 1.34i)14-s + (−2.53 − 6.12i)15-s + (−2.78 − 2.87i)16-s + (2.09 − 5.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.552 − 0.833i)2-s + (1.23 + 1.50i)3-s + (−0.389 + 0.920i)4-s + (−0.838 − 0.254i)5-s + (0.574 − 1.86i)6-s + (0.495 − 0.0985i)7-s + (0.982 − 0.183i)8-s + (−0.549 + 2.76i)9-s + (0.250 + 0.838i)10-s + (0.377 − 0.0371i)11-s + (−1.87 + 0.552i)12-s + (0.106 + 0.352i)13-s + (−0.355 − 0.358i)14-s + (−0.654 − 1.58i)15-s + (−0.696 − 0.717i)16-s + (0.509 − 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02688 + 0.269360i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02688 + 0.269360i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.781 + 1.17i)T \) |
| good | 3 | \( 1 + (-2.14 - 2.61i)T + (-0.585 + 2.94i)T^{2} \) |
| 5 | \( 1 + (1.87 + 0.568i)T + (4.15 + 2.77i)T^{2} \) |
| 7 | \( 1 + (-1.31 + 0.260i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.25 + 0.123i)T + (10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-0.385 - 1.27i)T + (-10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (-2.09 + 5.06i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.832 - 0.445i)T + (10.5 + 15.7i)T^{2} \) |
| 23 | \( 1 + (4.39 + 6.57i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.394 + 4.00i)T + (-28.4 - 5.65i)T^{2} \) |
| 31 | \( 1 + (-3.39 + 3.39i)T - 31iT^{2} \) |
| 37 | \( 1 + (-3.30 - 6.18i)T + (-20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (-0.369 + 0.246i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-3.34 + 4.07i)T + (-8.38 - 42.1i)T^{2} \) |
| 47 | \( 1 + (2.55 + 1.05i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.15 - 11.6i)T + (-51.9 + 10.3i)T^{2} \) |
| 59 | \( 1 + (-1.48 + 4.91i)T + (-49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (5.88 - 4.82i)T + (11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (-5.53 + 4.54i)T + (13.0 - 65.7i)T^{2} \) |
| 71 | \( 1 + (-0.650 - 3.26i)T + (-65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.353 - 0.0702i)T + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (2.08 - 0.864i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (3.28 - 6.13i)T + (-46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (0.923 - 1.38i)T + (-34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (0.135 - 0.135i)T - 97iT^{2} \) |
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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
−13.69563366669830840026922778030, −12.02469117627289773982886963415, −11.20317290141671219000847988479, −10.11117496980941326602449714868, −9.367594944588796524131257346187, −8.343486708031166853604292862248, −7.77652060929791551511759923983, −4.63896976499991138337878991622, −3.99411210051355505546236623544, −2.64852337642421032358265353293,
1.56112960814841965819628299788, 3.65075967435980890745730976615, 5.98228903478474297384171868857, 7.16109218198776642983918797321, 7.916352334997406101010689604557, 8.451300183580751130742642497822, 9.654099280574233203971453532995, 11.35194981270236916805636454782, 12.46483181672251027861886205636, 13.51836488185287741550866918809
