L(s) = 1 | + (0.161 − 1.40i)2-s − i·3-s + (−1.94 − 0.452i)4-s + (−1.55 − 1.60i)5-s + (−1.40 − 0.161i)6-s + (−0.299 + 2.62i)7-s + (−0.949 + 2.66i)8-s − 9-s + (−2.50 + 1.92i)10-s − 2.86i·11-s + (−0.452 + 1.94i)12-s − 4.36·13-s + (3.64 + 0.844i)14-s + (−1.60 + 1.55i)15-s + (3.59 + 1.76i)16-s − 5.54·17-s + ⋯ |
L(s) = 1 | + (0.113 − 0.993i)2-s − 0.577i·3-s + (−0.974 − 0.226i)4-s + (−0.694 − 0.719i)5-s + (−0.573 − 0.0657i)6-s + (−0.113 + 0.993i)7-s + (−0.335 + 0.941i)8-s − 0.333·9-s + (−0.793 + 0.608i)10-s − 0.864i·11-s + (−0.130 + 0.562i)12-s − 1.21·13-s + (0.974 + 0.225i)14-s + (−0.415 + 0.401i)15-s + (0.897 + 0.440i)16-s − 1.34·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.202119 + 0.313850i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.202119 + 0.313850i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.161 + 1.40i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.55 + 1.60i)T \) |
| 7 | \( 1 + (0.299 - 2.62i)T \) |
good | 11 | \( 1 + 2.86iT - 11T^{2} \) |
| 13 | \( 1 + 4.36T + 13T^{2} \) |
| 17 | \( 1 + 5.54T + 17T^{2} \) |
| 19 | \( 1 + 2.90T + 19T^{2} \) |
| 23 | \( 1 - 7.73T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 6.16T + 31T^{2} \) |
| 37 | \( 1 + 4.76iT - 37T^{2} \) |
| 41 | \( 1 + 1.97iT - 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 + 6.78iT - 47T^{2} \) |
| 53 | \( 1 + 7.68iT - 53T^{2} \) |
| 59 | \( 1 + 4.30T + 59T^{2} \) |
| 61 | \( 1 + 8.05iT - 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 - 0.551iT - 71T^{2} \) |
| 73 | \( 1 - 4.49T + 73T^{2} \) |
| 79 | \( 1 - 3.07iT - 79T^{2} \) |
| 83 | \( 1 - 7.45iT - 83T^{2} \) |
| 89 | \( 1 - 2.91iT - 89T^{2} \) |
| 97 | \( 1 + 9.73T + 97T^{2} \) |
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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
−10.98100358718989257447965007473, −9.550802621108359083140944708595, −8.754715746869346115381245025575, −8.251648667314958698445910110396, −6.81260062394886335774521888792, −5.43924917759424371995363433508, −4.65235892164193048691293618940, −3.21029407277686930847049326992, −2.06033813284864732608192244475, −0.22291323712856611520928588791,
3.03047925900694213573090690217, 4.38713287981987358360745345290, 4.73299669123828728573816896253, 6.52579690683321950215056295116, 7.08574240326300246108017333095, 7.88390683084866012828114777199, 9.040072603295120827430420329594, 9.982731413291038083331711671820, 10.69201197959674487591404579106, 11.76170578149230366465604311524