L(s) = 1 | + (−0.5 + 0.866i)2-s + (1.62 − 0.606i)3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.285 + 1.70i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (2.26 − 1.96i)9-s + 0.999·10-s + (1.28 − 2.22i)11-s + (−1.33 − 1.10i)12-s + (−3.45 − 5.99i)13-s + (0.499 + 0.866i)14-s + (−1.33 − 1.10i)15-s + (−0.5 + 0.866i)16-s − 4.52·17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.936 − 0.350i)3-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.116 + 0.697i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (0.754 − 0.655i)9-s + 0.316·10-s + (0.387 − 0.671i)11-s + (−0.385 − 0.318i)12-s + (−0.959 − 1.66i)13-s + (0.133 + 0.231i)14-s + (−0.345 − 0.284i)15-s + (−0.125 + 0.216i)16-s − 1.09·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26044 - 0.713249i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26044 - 0.713249i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1.62 + 0.606i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 11 | \( 1 + (-1.28 + 2.22i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.45 + 5.99i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 + 5.67T + 19T^{2} \) |
| 23 | \( 1 + (-4.03 - 6.98i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.21 + 5.56i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.57 - 2.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + (-4.38 - 7.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.60 + 9.70i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.143 + 0.249i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 + (-3.55 - 6.15i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.12 + 5.40i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.50 + 9.54i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 1.85T + 73T^{2} \) |
| 79 | \( 1 + (2.81 - 4.87i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.24 - 3.88i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + (6.70 - 11.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.22587745496201816610357998987, −9.318468012849885352996084977544, −8.536777963355961508118185131400, −7.916764990337484605502386966851, −7.16033104673138632486328592440, −6.14342286093539604243445624228, −4.92178780574405208936377907071, −3.83168734702416820479212658245, −2.49540626274668138113732062089, −0.807160995093518002325271732567,
2.03941468358774778276041679434, 2.64171364980915630447401414765, 4.35168725200362786567731039734, 4.46179509876493465374945709895, 6.66490032572331424546846017484, 7.24208153096729731825646767262, 8.545267723650309730635763860939, 8.972872544569662484342250423773, 9.779398591525323316018462244255, 10.67952161060059975304980907513