| L(s) = 1 | − 3.46i·5-s + i·7-s − 17.3·11-s − 1.73i·13-s − 6·17-s + 1.73·19-s + 30i·23-s + 13.0·25-s + 20.7i·29-s + 14i·31-s + 3.46·35-s − 19.0i·37-s + 48·41-s + 20.7·43-s − 66i·47-s + ⋯ |
| L(s) = 1 | − 0.692i·5-s + 0.142i·7-s − 1.57·11-s − 0.133i·13-s − 0.352·17-s + 0.0911·19-s + 1.30i·23-s + 0.520·25-s + 0.716i·29-s + 0.451i·31-s + 0.0989·35-s − 0.514i·37-s + 1.17·41-s + 0.483·43-s − 1.40i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.528818458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.528818458\) |
| \(L(2)\) |
|
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 | \( 1 + 3.46iT - 25T^{2} \) |
| 7 | \( 1 - iT - 49T^{2} \) |
| 11 | \( 1 + 17.3T + 121T^{2} \) |
| 13 | \( 1 + 1.73iT - 169T^{2} \) |
| 17 | \( 1 + 6T + 289T^{2} \) |
| 19 | \( 1 - 1.73T + 361T^{2} \) |
| 23 | \( 1 - 30iT - 529T^{2} \) |
| 29 | \( 1 - 20.7iT - 841T^{2} \) |
| 31 | \( 1 - 14iT - 961T^{2} \) |
| 37 | \( 1 + 19.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 48T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 66iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 43.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 60.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 48iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 49T + 5.32e3T^{2} \) |
| 79 | \( 1 - 83iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 13.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 66T + 7.92e3T^{2} \) |
| 97 | \( 1 - 107T + 9.40e3T^{2} \) |
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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
−9.086419069898397968225685902144, −8.418151302141121822626981952021, −7.64572552576319639987773693523, −6.92495149049612364119223565588, −5.56637800696600782595514513315, −5.30252508903341195663936554887, −4.26369006228061743963224115345, −3.12417863571975888186224830844, −2.13184933974086871220873150328, −0.78573954417871106432176824626,
0.56179982159950396696951562158, 2.32923136887219548673528851126, 2.86370323806967105917714288126, 4.13247335340617096123584991567, 4.97685686849470007496301557249, 5.95208752329055444344984101525, 6.71937992556930307444493328060, 7.58070136735076631090333280202, 8.165411016966105330582376776850, 9.111865925520579783681401383775
