| L(s) = 1 | + 1.41i·2-s + 1.48i·3-s + 1.98·4-s − 2.10·6-s − 2.50·7-s + 8.49i·8-s + 6.80·9-s + 16.7·11-s + 2.94i·12-s − 12.8i·13-s − 3.55i·14-s − 4.09·16-s + 10.1·17-s + 9.65i·18-s + (2.53 + 18.8i)19-s + ⋯ |
| L(s) = 1 | + 0.709i·2-s + 0.493i·3-s + 0.497·4-s − 0.350·6-s − 0.357·7-s + 1.06i·8-s + 0.756·9-s + 1.52·11-s + 0.245i·12-s − 0.991i·13-s − 0.253i·14-s − 0.255·16-s + 0.595·17-s + 0.536i·18-s + (0.133 + 0.991i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.309539847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.309539847\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (-2.53 - 18.8i)T \) |
| good | 2 | \( 1 - 1.41iT - 4T^{2} \) |
| 3 | \( 1 - 1.48iT - 9T^{2} \) |
| 7 | \( 1 + 2.50T + 49T^{2} \) |
| 11 | \( 1 - 16.7T + 121T^{2} \) |
| 13 | \( 1 + 12.8iT - 169T^{2} \) |
| 17 | \( 1 - 10.1T + 289T^{2} \) |
| 23 | \( 1 + 18.7T + 529T^{2} \) |
| 29 | \( 1 - 30.4iT - 841T^{2} \) |
| 31 | \( 1 + 56.8iT - 961T^{2} \) |
| 37 | \( 1 - 19.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 15.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 68.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 33.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.77iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 41.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 80.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 73.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 119. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 66.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 56.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 124.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 9.31iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 7.61iT - 9.40e3T^{2} \) |
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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
−10.94063445745945127464373932175, −10.05098041281804465191706971216, −9.338664145803379935219783817875, −8.105212316334153995859181934607, −7.37112669342230344991513957444, −6.33777603630185057225554315113, −5.67023880404492075377705958751, −4.29608581522551818593735712266, −3.25063006375562825384432279587, −1.51702476068134777396681420290,
1.09688405466874915297776229672, 2.06859536402877141309207289369, 3.49553727385813274139856401261, 4.41972396304761580938689480332, 6.26698176355743601531751946106, 6.76198850376472116346335577111, 7.59762892961199695408569969006, 9.096984087359679306573719038190, 9.692496585205142775044147854988, 10.65944109802634536344560155855
