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 + (−2.00 − 2.44i)10-s − 2.86i·11-s + (−0.452 + 1.94i)12-s − 4.36·13-s + (−3.74 − 0.00187i)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 + (−0.635 − 0.772i)10-s − 0.864i·11-s + (−0.130 + 0.562i)12-s − 1.21·13-s + (−0.999 − 0.000501i)14-s + (0.415 + 0.401i)15-s + (0.897 + 0.440i)16-s − 1.34·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0671481 - 0.187464i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0671481 - 0.187464i\) |
\(L(\frac{3}{2})\) |
|
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
−11.82937364692775395504596434601, −10.93408817091724688780696699151, −9.663544887308338270055713695238, −8.740779054234999841170726332664, −7.911084552544889733443283293671, −7.17596516094259033373051940173, −6.22412110843773604635585759128, −5.37099993628465202804213922359, −3.98256131123009538343438132514, −2.48277174302074533205482946640,
0.12700169012267913466934760631, 2.09378263475968718648306799544, 3.84241803021469571568842555066, 4.37024152003525106290687633004, 5.27332293355576686046820558637, 7.29151618929348529786786889117, 7.975669832280458590602299382122, 9.245327566996271077458927265964, 9.702845772354781643905998583792, 10.74058002793240086780111217619