L(s) = 1 | + (1.61 + 1.17i)2-s + (1.23 + 3.80i)4-s − 2.23·5-s + 8.50i·7-s + (−2.47 + 7.60i)8-s + (−3.61 − 2.62i)10-s + 1.79i·11-s + 0.472·13-s + (−10 + 13.7i)14-s + (−12.9 + 9.40i)16-s + 23.8·17-s + 9.40i·19-s + (−2.76 − 8.50i)20-s + (−2.11 + 2.90i)22-s − 16.1i·23-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s − 0.447·5-s + 1.21i·7-s + (−0.309 + 0.951i)8-s + (−0.361 − 0.262i)10-s + 0.163i·11-s + 0.0363·13-s + (−0.714 + 0.983i)14-s + (−0.809 + 0.587i)16-s + 1.40·17-s + 0.494i·19-s + (−0.138 − 0.425i)20-s + (−0.0959 + 0.132i)22-s − 0.700i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.951i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.22293 + 1.68321i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22293 + 1.68321i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.61 - 1.17i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
good | 7 | \( 1 - 8.50iT - 49T^{2} \) |
| 11 | \( 1 - 1.79iT - 121T^{2} \) |
| 13 | \( 1 - 0.472T + 169T^{2} \) |
| 17 | \( 1 - 23.8T + 289T^{2} \) |
| 19 | \( 1 - 9.40iT - 361T^{2} \) |
| 23 | \( 1 + 16.1iT - 529T^{2} \) |
| 29 | \( 1 + 6.94T + 841T^{2} \) |
| 31 | \( 1 + 47.4iT - 961T^{2} \) |
| 37 | \( 1 - 26.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 41.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 2.00iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 21.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 73.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 26.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 88.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 39.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 21.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 39.1T + 9.40e3T^{2} \) |
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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
−12.51666769876979165498290159691, −12.14531160836601808451197827819, −11.11600710502212103317796420764, −9.552779596615961171881707755653, −8.357881654070276076538388048451, −7.57315541498131009025042517263, −6.17013446208395822952469741981, −5.34113970026599315939787309665, −3.96119809043752482827936122196, −2.57717710395715256225166291396,
1.06184625448812135752651233757, 3.18716029263427950265925992322, 4.18585968978765290348821271261, 5.41885683525770648456837903213, 6.82252430635011132095208689330, 7.79653319208947233388284032214, 9.451583086160774726249118546747, 10.44816809918389292977110307322, 11.18489475553934111227339603006, 12.17935458242749947502053338363