| L(s) = 1 | − 3.65i·5-s + 9.65i·7-s + 18.4·11-s − 11.6i·13-s − 9.31·17-s − 15.1·19-s − 22.3i·23-s + 11.6·25-s − 28.3i·29-s + 45.2i·31-s + 35.3·35-s − 49.5i·37-s − 20.6·41-s + 46.0·43-s − 12.6i·47-s + ⋯ |
| L(s) = 1 | − 0.731i·5-s + 1.37i·7-s + 1.68·11-s − 0.896i·13-s − 0.547·17-s − 0.798·19-s − 0.971i·23-s + 0.465·25-s − 0.977i·29-s + 1.45i·31-s + 1.00·35-s − 1.34i·37-s − 0.503·41-s + 1.07·43-s − 0.269i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.105944552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.105944552\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.65iT - 25T^{2} \) |
| 7 | \( 1 - 9.65iT - 49T^{2} \) |
| 11 | \( 1 - 18.4T + 121T^{2} \) |
| 13 | \( 1 + 11.6iT - 169T^{2} \) |
| 17 | \( 1 + 9.31T + 289T^{2} \) |
| 19 | \( 1 + 15.1T + 361T^{2} \) |
| 23 | \( 1 + 22.3iT - 529T^{2} \) |
| 29 | \( 1 + 28.3iT - 841T^{2} \) |
| 31 | \( 1 - 45.2iT - 961T^{2} \) |
| 37 | \( 1 + 49.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 20.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 46.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 12.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 27.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 9.11T + 3.48e3T^{2} \) |
| 61 | \( 1 - 113. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 45.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 16.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 11.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 70.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 94.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 110.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 25.4T + 9.40e3T^{2} \) |
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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
−8.806116818131483418180681631135, −8.333567050007827466786357334775, −7.07808387851279568296149149248, −6.27338022436355372863160583089, −5.64997423126542005130067398727, −4.73891294917769706992611122432, −3.95946252006761789603025741066, −2.77113432146537968309955450296, −1.83091508349165549872240067459, −0.61567132804974037107282667094,
0.983788732941097694282850669158, 1.96959879690905122218169100837, 3.36414892527967407902192693925, 4.03269808339701085942150351136, 4.64386348120246511166430196118, 6.10982355959288195337422028926, 6.80086096085672194404139190479, 7.05866670339591497807047120773, 8.082608952407256871232473410941, 9.076736289859262565329428253555
