L(s) = 1 | + (0.780 − 1.17i)2-s + (0.848 − 1.51i)3-s + (−0.780 − 1.84i)4-s + (−1.11 − 2.17i)6-s + 3.02·7-s + (−2.78 − 0.516i)8-s + (−1.56 − 2.56i)9-s − 1.32·11-s + (−3.44 − 0.382i)12-s + 5.12i·13-s + (2.35 − 3.56i)14-s + (−2.78 + 2.87i)16-s + 2·17-s + (−4.23 − 0.158i)18-s − 1.32i·19-s + ⋯ |
L(s) = 1 | + (0.552 − 0.833i)2-s + (0.489 − 0.871i)3-s + (−0.390 − 0.920i)4-s + (−0.456 − 0.889i)6-s + 1.14·7-s + (−0.983 − 0.182i)8-s + (−0.520 − 0.853i)9-s − 0.399·11-s + (−0.993 − 0.110i)12-s + 1.42i·13-s + (0.630 − 0.951i)14-s + (−0.695 + 0.718i)16-s + 0.485·17-s + (−0.999 − 0.0374i)18-s − 0.303i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.935972 - 1.72036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.935972 - 1.72036i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.780 + 1.17i)T \) |
| 3 | \( 1 + (-0.848 + 1.51i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 - 5.12iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 1.32iT - 19T^{2} \) |
| 23 | \( 1 - 0.371iT - 23T^{2} \) |
| 29 | \( 1 - 3.12iT - 29T^{2} \) |
| 31 | \( 1 + 4.71iT - 31T^{2} \) |
| 37 | \( 1 - 5.12iT - 37T^{2} \) |
| 41 | \( 1 - 1.12iT - 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 + 3.02iT - 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 4.34T + 67T^{2} \) |
| 71 | \( 1 + 3.39T + 71T^{2} \) |
| 73 | \( 1 + 8.24iT - 73T^{2} \) |
| 79 | \( 1 + 8.10iT - 79T^{2} \) |
| 83 | \( 1 - 15.1iT - 83T^{2} \) |
| 89 | \( 1 - 10.2iT - 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−11.64545156259980234049455684980, −10.82953525586046573841663942616, −9.510933335527652411002013679363, −8.670882747499787560961316149144, −7.60026658526605991785112430373, −6.43149351347120934594536659299, −5.18914893423673053298608777410, −4.02174634222205364399853565857, −2.52278045069656088771437042037, −1.43874862660615158718974852376,
2.76759494464878217451822666490, 4.00553694726935926490254029443, 5.08684597800783301178237091243, 5.73495486432616189016803744149, 7.55970278148688590662654767500, 8.071291287140766374187174801019, 8.942636608624617390954682909013, 10.19652730911956519859690662475, 11.04934157067206754370528870570, 12.20795465951569178851898124527