L(s) = 1 | + (−0.587 − 0.809i)2-s + (3.08 + 1.00i)3-s + (−0.309 + 0.951i)4-s + (1.12 − 1.93i)5-s + (−1.00 − 3.08i)6-s + i·7-s + (0.951 − 0.309i)8-s + (6.09 + 4.42i)9-s + (−2.22 + 0.226i)10-s + (−2.11 + 1.53i)11-s + (−1.90 + 2.62i)12-s + (1.40 − 1.92i)13-s + (0.809 − 0.587i)14-s + (5.40 − 4.83i)15-s + (−0.809 − 0.587i)16-s + (−6.80 + 2.21i)17-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.572i)2-s + (1.78 + 0.578i)3-s + (−0.154 + 0.475i)4-s + (0.502 − 0.864i)5-s + (−0.409 − 1.25i)6-s + 0.377i·7-s + (0.336 − 0.109i)8-s + (2.03 + 1.47i)9-s + (−0.703 + 0.0715i)10-s + (−0.636 + 0.462i)11-s + (−0.550 + 0.757i)12-s + (0.388 − 0.534i)13-s + (0.216 − 0.157i)14-s + (1.39 − 1.24i)15-s + (−0.202 − 0.146i)16-s + (−1.65 + 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.93004 - 0.291567i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.93004 - 0.291567i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 + (-1.12 + 1.93i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (-3.08 - 1.00i)T + (2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (2.11 - 1.53i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.40 + 1.92i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (6.80 - 2.21i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.24 + 6.90i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.69 - 3.70i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.830 - 2.55i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.978 - 3.01i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.27 + 5.87i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (2.86 + 2.08i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.58iT - 43T^{2} \) |
| 47 | \( 1 + (6.84 + 2.22i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.57 + 0.836i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-9.59 - 6.97i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.723 + 0.526i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.998 - 0.324i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.940 - 2.89i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (6.68 + 9.19i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (3.82 - 11.7i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.21 + 2.01i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (9.66 - 7.01i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (11.0 + 3.58i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.12294549227878883638691125049, −10.26386556650586322368295135841, −9.334587575735974669035557732085, −8.811748807754492281009838735197, −8.284670292046002277027553871488, −7.06172895564438197604447429719, −5.07230159219755408164033389244, −4.16232604990969633921559284177, −2.79089525529366297385590526104, −1.93653036535563758149566268924,
1.86522503094403581965342037378, 2.93726269398254005404828106656, 4.25621374312239673418100835254, 6.29067645911187448128323942447, 6.88302982672634342598800243218, 7.915712306628079356473674658040, 8.521525566111867464204144891323, 9.493943925178395312671102983048, 10.22987049222655450104128337042, 11.31144749993075799355515062648