L(s) = 1 | + (−0.781 − 0.623i)2-s + (0.864 − 1.79i)3-s + (0.222 + 0.974i)4-s + (1.67 + 1.48i)5-s + (−1.79 + 0.864i)6-s + (2.09 − 1.61i)7-s + (0.433 − 0.900i)8-s + (−0.605 − 0.759i)9-s + (−0.383 − 2.20i)10-s + (−0.709 + 0.890i)11-s + (1.94 + 0.443i)12-s + (1.60 + 1.27i)13-s + (−2.64 − 0.0449i)14-s + (4.10 − 1.72i)15-s + (−0.900 + 0.433i)16-s + (−0.405 − 0.0925i)17-s + ⋯ |
L(s) = 1 | + (−0.552 − 0.440i)2-s + (0.499 − 1.03i)3-s + (0.111 + 0.487i)4-s + (0.748 + 0.663i)5-s + (−0.733 + 0.352i)6-s + (0.792 − 0.610i)7-s + (0.153 − 0.318i)8-s + (−0.201 − 0.253i)9-s + (−0.121 − 0.696i)10-s + (−0.214 + 0.268i)11-s + (0.560 + 0.128i)12-s + (0.444 + 0.354i)13-s + (−0.707 − 0.0120i)14-s + (1.06 − 0.444i)15-s + (−0.225 + 0.108i)16-s + (−0.0983 − 0.0224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35335 - 0.840817i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35335 - 0.840817i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.781 + 0.623i)T \) |
| 5 | \( 1 + (-1.67 - 1.48i)T \) |
| 7 | \( 1 + (-2.09 + 1.61i)T \) |
good | 3 | \( 1 + (-0.864 + 1.79i)T + (-1.87 - 2.34i)T^{2} \) |
| 11 | \( 1 + (0.709 - 0.890i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.60 - 1.27i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (0.405 + 0.0925i)T + (15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 + (0.412 - 0.0942i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-0.276 + 1.20i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + 2.98T + 31T^{2} \) |
| 37 | \( 1 + (2.53 + 0.577i)T + (33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (8.98 + 4.32i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (4.11 + 8.53i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (0.165 + 0.132i)T + (10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (12.9 - 2.95i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-3.09 + 1.49i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (0.440 - 1.93i)T + (-54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 + (-1.57 - 6.91i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.40 - 3.50i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + (4.28 - 3.41i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-10.4 - 13.0i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + 4.55iT - 97T^{2} \) |
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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
−10.72828107432990021340182651238, −10.02314461848118253683660360497, −8.962619080993275268362732211306, −8.020397341252137963425320963874, −7.27385796936288881419009183431, −6.62510087606038831532450400905, −5.15104194077005023107915411753, −3.54996447265413219378055566640, −2.22962555760614314530902856984, −1.43644688910069177080755349174,
1.55844919212735858370637510150, 3.15400552788611562603602337153, 4.71486269762173148651909162255, 5.33543023199160535174960798246, 6.38861419903231691456557137571, 7.922220716016630205355711487327, 8.594322607394932829828148212988, 9.279548847121989374815548115249, 9.946459491144859371200704204040, 10.79776465080201794161571097854