L(s) = 1 | + 2.30·2-s + (1.08 − 1.87i)3-s + 3.30·4-s + (1.08 − 1.87i)5-s + (2.49 − 4.32i)6-s + 3.00·8-s + (−0.848 − 1.46i)9-s + (2.49 − 4.32i)10-s + (−2.45 + 4.25i)11-s + (3.57 − 6.19i)12-s + (1.41 + 3.31i)13-s + (−2.34 − 4.06i)15-s + 0.302·16-s − 7.15·17-s + (−1.95 − 3.38i)18-s + (1.08 + 1.87i)19-s + ⋯ |
L(s) = 1 | + 1.62·2-s + (0.625 − 1.08i)3-s + 1.65·4-s + (0.484 − 0.839i)5-s + (1.01 − 1.76i)6-s + 1.06·8-s + (−0.282 − 0.489i)9-s + (0.789 − 1.36i)10-s + (−0.739 + 1.28i)11-s + (1.03 − 1.78i)12-s + (0.391 + 0.920i)13-s + (−0.606 − 1.05i)15-s + 0.0756·16-s − 1.73·17-s + (−0.460 − 0.797i)18-s + (0.248 + 0.430i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.82001 - 2.15439i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.82001 - 2.15439i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-1.41 - 3.31i)T \) |
good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 + (-1.08 + 1.87i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.08 + 1.87i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.45 - 4.25i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 7.15T + 17T^{2} \) |
| 19 | \( 1 + (-1.08 - 1.87i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 0.605T + 23T^{2} \) |
| 29 | \( 1 + (-1.15 - 1.99i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.57 + 6.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.60T + 37T^{2} \) |
| 41 | \( 1 + (4.99 + 8.64i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.25 + 10.8i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.755 - 1.30i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.19 + 2.07i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + (-2.16 - 3.75i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.16 + 3.75i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.30 + 5.72i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 6.50T + 89T^{2} \) |
| 97 | \( 1 + (6.83 - 11.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.75473291185574202753637050055, −9.386933651374478029644813280958, −8.623695382880687470022260417974, −7.37894274828187954901925717413, −6.82314430202803930276524857900, −5.77701547609468016565582207445, −4.80106129777003943672963717987, −4.03969752380130787483781398091, −2.37293705125911309265108419578, −1.89143819262262477746876585991,
2.76141597478575281667236073436, 3.00539110293087807965607387180, 4.14096778828151628742803287955, 5.01768060149473042878637678907, 6.01952698184159515872301130218, 6.67405848220992671874603117141, 8.121807115043959697203758334674, 9.056610307734375115193419430546, 10.13206185344224617196924687772, 11.00152284190360587136339955208