L(s) = 1 | + (−1.15 + 0.809i)2-s + (−0.147 − 0.551i)3-s + (0.690 − 1.87i)4-s + (1.15 + 1.91i)5-s + (0.618 + 0.520i)6-s + (−0.735 − 2.54i)7-s + (0.719 + 2.73i)8-s + (2.31 − 1.33i)9-s + (−2.88 − 1.29i)10-s + (3.42 + 1.98i)11-s + (−1.13 − 0.103i)12-s + (2.04 + 2.04i)13-s + (2.90 + 2.35i)14-s + (0.888 − 0.918i)15-s + (−3.04 − 2.59i)16-s + (−0.155 − 0.580i)17-s + ⋯ |
L(s) = 1 | + (−0.820 + 0.572i)2-s + (−0.0853 − 0.318i)3-s + (0.345 − 0.938i)4-s + (0.514 + 0.857i)5-s + (0.252 + 0.212i)6-s + (−0.277 − 0.960i)7-s + (0.254 + 0.967i)8-s + (0.771 − 0.445i)9-s + (−0.912 − 0.408i)10-s + (1.03 + 0.597i)11-s + (−0.328 − 0.0297i)12-s + (0.568 + 0.568i)13-s + (0.777 + 0.628i)14-s + (0.229 − 0.237i)15-s + (−0.761 − 0.647i)16-s + (−0.0377 − 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.835102 + 0.159080i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.835102 + 0.159080i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.15 - 0.809i)T \) |
| 5 | \( 1 + (-1.15 - 1.91i)T \) |
| 7 | \( 1 + (0.735 + 2.54i)T \) |
good | 3 | \( 1 + (0.147 + 0.551i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-3.42 - 1.98i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.04 - 2.04i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.155 + 0.580i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.00 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.39 + 0.640i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 7.25iT - 29T^{2} \) |
| 31 | \( 1 + (2.83 + 1.63i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (9.29 + 2.49i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 3.35T + 41T^{2} \) |
| 43 | \( 1 + (-3.31 + 3.31i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.24 - 12.1i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.90 + 0.779i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (7.41 - 12.8i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.11 + 3.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.67 + 0.984i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 4.86iT - 71T^{2} \) |
| 73 | \( 1 + (-9.44 + 2.53i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (3.45 + 5.98i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.46 + 5.46i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.35 - 1.93i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.11 + 1.11i)T - 97iT^{2} \) |
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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
−13.58786427397432330314897129960, −12.04228959687087367104248981111, −10.88263109995918165512010883305, −9.948232506454421531735362185963, −9.292157504875942282602183970173, −7.59339276159690433967471730242, −6.82712679783044753393805615963, −6.14606825656098142790279732320, −4.03596012374073146411046609334, −1.62735153914142671557160614790,
1.64053576479233959535615754039, 3.56896428883748279684767027922, 5.24411745706308979057425411373, 6.67176918321184019834597616346, 8.338207348422287199667315929431, 9.023697558983891458871762523978, 9.875111183411218881510707681784, 10.93614647278149834652689481493, 12.08043361126402327491771099485, 12.80030736227591634776434416765