L(s) = 1 | + (1.14 + 1.97i)2-s + (−0.610 − 1.05i)3-s + (−1.61 + 2.79i)4-s + 2.50·5-s + (1.39 − 2.41i)6-s + (−2.25 + 3.90i)7-s − 2.79·8-s + (0.753 − 1.30i)9-s + (2.86 + 4.96i)10-s + (−0.5 − 0.866i)11-s + 3.93·12-s + (−2.5 − 2.59i)13-s − 10.2·14-s + (−1.53 − 2.65i)15-s + (0.0316 + 0.0547i)16-s + (1.61 − 2.79i)17-s + ⋯ |
L(s) = 1 | + (0.807 + 1.39i)2-s + (−0.352 − 0.610i)3-s + (−0.805 + 1.39i)4-s + 1.12·5-s + (0.569 − 0.987i)6-s + (−0.851 + 1.47i)7-s − 0.987·8-s + (0.251 − 0.435i)9-s + (0.905 + 1.56i)10-s + (−0.150 − 0.261i)11-s + 1.13·12-s + (−0.693 − 0.720i)13-s − 2.75·14-s + (−0.395 − 0.684i)15-s + (0.00790 + 0.0136i)16-s + (0.390 − 0.676i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09989 + 1.08588i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09989 + 1.08588i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (2.5 + 2.59i)T \) |
good | 2 | \( 1 + (-1.14 - 1.97i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.610 + 1.05i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 2.50T + 5T^{2} \) |
| 7 | \( 1 + (2.25 - 3.90i)T + (-3.5 - 6.06i)T^{2} \) |
| 17 | \( 1 + (-1.61 + 2.79i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.17 + 2.03i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.89 + 5.01i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.536i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.01T + 31T^{2} \) |
| 37 | \( 1 + (2.53 + 4.38i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.18 - 7.24i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.54 - 9.60i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + 4.06T + 53T^{2} \) |
| 59 | \( 1 + (5.53 - 9.58i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.0722 + 0.125i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.36 + 7.55i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.857 + 1.48i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 1.87T + 79T^{2} \) |
| 83 | \( 1 + 5.38T + 83T^{2} \) |
| 89 | \( 1 + (-4.25 - 7.36i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0562 - 0.0974i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.36756094694110395029723012283, −12.69369532076517906283900919291, −12.01171249746395710739633641265, −9.989952929796305960170253565883, −9.049948998020525886885825402569, −7.68439637528586180533431846165, −6.36935453401415002536656161307, −6.03268270470242332116633811549, −5.03267913168695364035468188565, −2.78940096442065197352792970533,
1.84600934585149180624243899777, 3.61677235568673405868551834998, 4.59405422214158316108982665357, 5.79953328225383607209502418739, 7.32629102233358142052643440996, 9.716173520272395531272323110142, 10.04276939467740251449158686674, 10.62565807290168967071578380746, 11.88111653171697433850913843957, 12.92078433401637180533982538053