| L(s) = 1 | − 6.81i·5-s − 2.64·7-s + 1.61i·11-s − 0.520·13-s − 26.9i·17-s − 17.8·19-s + 12.1i·23-s − 21.4·25-s + 47.9i·29-s − 46.8·31-s + 18.0i·35-s + 1.85·37-s − 41.6i·41-s + 2.33·43-s + 91.9i·47-s + ⋯ |
| L(s) = 1 | − 1.36i·5-s − 0.377·7-s + 0.146i·11-s − 0.0400·13-s − 1.58i·17-s − 0.941·19-s + 0.529i·23-s − 0.856·25-s + 1.65i·29-s − 1.51·31-s + 0.515i·35-s + 0.0502·37-s − 1.01i·41-s + 0.0542·43-s + 1.95i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8527119786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8527119786\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 + 6.81iT - 25T^{2} \) |
| 11 | \( 1 - 1.61iT - 121T^{2} \) |
| 13 | \( 1 + 0.520T + 169T^{2} \) |
| 17 | \( 1 + 26.9iT - 289T^{2} \) |
| 19 | \( 1 + 17.8T + 361T^{2} \) |
| 23 | \( 1 - 12.1iT - 529T^{2} \) |
| 29 | \( 1 - 47.9iT - 841T^{2} \) |
| 31 | \( 1 + 46.8T + 961T^{2} \) |
| 37 | \( 1 - 1.85T + 1.36e3T^{2} \) |
| 41 | \( 1 + 41.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 2.33T + 1.84e3T^{2} \) |
| 47 | \( 1 - 91.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 30.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 72.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 32.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 45.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 111. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 99.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 3.78T + 6.24e3T^{2} \) |
| 83 | \( 1 - 35.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 48.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 12.1T + 9.40e3T^{2} \) |
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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
−8.573693898939185251827622590268, −7.57735115906348485831095141417, −7.08494251590745180526191131094, −6.05548478827958901226153789383, −5.25180605034781115254145346330, −4.76411054172589838622896407901, −3.89005544947548589393675426384, −2.91158321854376864712781853379, −1.79525009611579286607102610988, −0.821059685828650322005875945311,
0.21405545560697388361896135544, 1.88977775590849504492163033965, 2.56794350945189203199078838283, 3.62708936241974090961481363883, 4.03204423424802310951373667713, 5.31396846713010277970424197410, 6.32615013820378525233159265086, 6.44412745665804542305932841179, 7.36472835664566361206483813810, 8.152086420720040141224657823553
