L(s) = 1 | − 1.73i·3-s + 11.2i·7-s − 2.99·9-s − 4.28i·11-s + 21.4·13-s − 25.4·17-s − 22.4i·19-s + 19.4·21-s + 37.9i·23-s + 5.19i·27-s + 1.41·29-s + 19.1i·31-s − 7.41·33-s + 20.2·37-s − 37.0i·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 1.60i·7-s − 0.333·9-s − 0.389i·11-s + 1.64·13-s − 1.49·17-s − 1.18i·19-s + 0.924·21-s + 1.64i·23-s + 0.192i·27-s + 0.0488·29-s + 0.617i·31-s − 0.224·33-s + 0.547·37-s − 0.951i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.354481988\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.354481988\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 11.2iT - 49T^{2} \) |
| 11 | \( 1 + 4.28iT - 121T^{2} \) |
| 13 | \( 1 - 21.4T + 169T^{2} \) |
| 17 | \( 1 + 25.4T + 289T^{2} \) |
| 19 | \( 1 + 22.4iT - 361T^{2} \) |
| 23 | \( 1 - 37.9iT - 529T^{2} \) |
| 29 | \( 1 - 1.41T + 841T^{2} \) |
| 31 | \( 1 - 19.1iT - 961T^{2} \) |
| 37 | \( 1 - 20.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30T + 1.68e3T^{2} \) |
| 43 | \( 1 + 24.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 70.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 64.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 88.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 66.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 36.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 133. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 28.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 60.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 4.90iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 32.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 14T + 9.40e3T^{2} \) |
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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
−9.392253588538735702742924150310, −8.838624357427555369694370377802, −8.392159361759177853159155377961, −7.23347539170887584947928530179, −6.24810246652016911930101309945, −5.81282475086683097820244105472, −4.75127238025683838965803392255, −3.37543022029526349019377703934, −2.45350053073465611441154772048, −1.35071205600456593276720958549,
0.41027379479192142026813010121, 1.79928678982369594079512036165, 3.41198100137483179611167033348, 4.13415676555331540296488734531, 4.74784196230258015341165947757, 6.23010327664442194205944015365, 6.66750936527277954504897570143, 7.86660684660760960118443880324, 8.486987142113403014532332298585, 9.445451463669136200036898966939