L(s) = 1 | − i·2-s + (−1.72 + 0.168i)3-s − 4-s + (−1.95 − 1.08i)5-s + (0.168 + 1.72i)6-s + (0.831 − 2.51i)7-s + i·8-s + (2.94 − 0.580i)9-s + (−1.08 + 1.95i)10-s + (0.171 − 0.296i)11-s + (1.72 − 0.168i)12-s + (−4.33 − 2.50i)13-s + (−2.51 − 0.831i)14-s + (3.55 + 1.53i)15-s + 16-s + (−0.981 + 0.566i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.995 + 0.0972i)3-s − 0.5·4-s + (−0.875 − 0.483i)5-s + (0.0687 + 0.703i)6-s + (0.314 − 0.949i)7-s + 0.353i·8-s + (0.981 − 0.193i)9-s + (−0.341 + 0.618i)10-s + (0.0516 − 0.0895i)11-s + (0.497 − 0.0486i)12-s + (−1.20 − 0.694i)13-s + (−0.671 − 0.222i)14-s + (0.918 + 0.396i)15-s + 0.250·16-s + (−0.238 + 0.137i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0111896 + 0.0127718i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0111896 + 0.0127718i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.72 - 0.168i)T \) |
| 5 | \( 1 + (1.95 + 1.08i)T \) |
| 7 | \( 1 + (-0.831 + 2.51i)T \) |
good | 11 | \( 1 + (-0.171 + 0.296i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.33 + 2.50i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.981 - 0.566i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0114 - 0.0197i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.65 - 3.83i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.68 - 4.64i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + (-6.66 - 3.84i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.52 - 4.36i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.27 - 1.89i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 5.35iT - 47T^{2} \) |
| 53 | \( 1 + (3.77 - 2.18i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 6.85T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.4iT - 67T^{2} \) |
| 71 | \( 1 + 7.78T + 71T^{2} \) |
| 73 | \( 1 + (-11.7 + 6.80i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 5.01T + 79T^{2} \) |
| 83 | \( 1 + (-0.917 + 0.529i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.36 + 5.83i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.3 + 8.86i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.19993390950671798217500131213, −9.438706692991954322706965927757, −7.997561675743525686712480406811, −7.53243303654617689822938763444, −6.25141491382240811290259209664, −4.88945721707226762875356143593, −4.49634733007173111892003706192, −3.33195144364358900217895203674, −1.34025532817720711666892430389, −0.01133462689507369654337990317,
2.34428622437080960112148804873, 4.18958939269574228177525427831, 4.84607487448758422336482116921, 5.98186741513860196934398123558, 6.71203778990275692046957406508, 7.57771448198055217921631974632, 8.350794107756931201457553841754, 9.529741808439869025991552358495, 10.35407250067202130711256686141, 11.43494762122561897956431476082