| L(s) = 1 | + (1.26 + 0.634i)2-s + (−1.41 − 0.993i)3-s + (1.19 + 1.60i)4-s + (−0.160 + 0.0486i)5-s + (−1.16 − 2.15i)6-s + (−0.313 + 1.57i)7-s + (0.493 + 2.78i)8-s + (1.02 + 2.81i)9-s + (−0.233 − 0.0402i)10-s + (−0.127 + 1.29i)11-s + (−0.102 − 3.46i)12-s + (5.48 + 1.66i)13-s + (−1.39 + 1.79i)14-s + (0.276 + 0.0903i)15-s + (−1.14 + 3.83i)16-s + (3.05 − 1.26i)17-s + ⋯ |
| L(s) = 1 | + (0.893 + 0.448i)2-s + (−0.819 − 0.573i)3-s + (0.597 + 0.801i)4-s + (−0.0717 + 0.0217i)5-s + (−0.474 − 0.880i)6-s + (−0.118 + 0.596i)7-s + (0.174 + 0.984i)8-s + (0.341 + 0.939i)9-s + (−0.0738 − 0.0127i)10-s + (−0.0385 + 0.391i)11-s + (−0.0296 − 0.999i)12-s + (1.52 + 0.461i)13-s + (−0.373 + 0.479i)14-s + (0.0712 + 0.0233i)15-s + (−0.285 + 0.958i)16-s + (0.741 − 0.307i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52122 + 0.923427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52122 + 0.923427i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 - 0.634i)T \) |
| 3 | \( 1 + (1.41 + 0.993i)T \) |
| good | 5 | \( 1 + (0.160 - 0.0486i)T + (4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (0.313 - 1.57i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (0.127 - 1.29i)T + (-10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-5.48 - 1.66i)T + (10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (-3.05 + 1.26i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.517 + 0.276i)T + (10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (0.750 - 1.12i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.813 + 8.26i)T + (-28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (-0.245 + 0.245i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.71 - 0.918i)T + (20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (3.17 - 4.74i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (3.82 + 4.66i)T + (-8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (8.52 - 3.53i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.372 + 3.78i)T + (-51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (-3.41 + 1.03i)T + (49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-7.54 + 9.19i)T + (-11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (9.47 + 7.77i)T + (13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (-1.05 + 5.31i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-6.82 + 1.35i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-4.45 + 10.7i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-7.46 + 3.98i)T + (46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (7.70 - 5.14i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (5.99 + 5.99i)T + 97iT^{2} \) |
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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
−11.66397591762559160195625776075, −11.11773750178005842187959231227, −9.750462077801171912964131497331, −8.333201167527589543485176184133, −7.55077309147038514180858920804, −6.42099833561159914269730693940, −5.86945815209979366116670292708, −4.84703721391270279142107349626, −3.56184945375325118993073188995, −1.91461946280196948565325483160,
1.12054804178228283399331111205, 3.38872284118762978022639481912, 4.03792337277692224706553372418, 5.33344158874059727078243653535, 6.04115451630230961054410170495, 7.00231428966625578856262035343, 8.492991720041084096047304219337, 9.874486774717290482313750099161, 10.52398801427042455943560655542, 11.16584386198039192192504542321
