| L(s) = 1 | + 3.34i·5-s + 5.01·7-s − 7.85i·11-s − 14.8·13-s − 21.2i·17-s + 24.0·19-s + 21.0i·23-s + 13.8·25-s − 31.4i·29-s + 4.08·31-s + 16.7i·35-s − 41.3·37-s + 4.27i·41-s − 16.9·43-s + 66.0i·47-s + ⋯ |
| L(s) = 1 | + 0.668i·5-s + 0.716·7-s − 0.714i·11-s − 1.14·13-s − 1.25i·17-s + 1.26·19-s + 0.915i·23-s + 0.552·25-s − 1.08i·29-s + 0.131·31-s + 0.479i·35-s − 1.11·37-s + 0.104i·41-s − 0.394·43-s + 1.40i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.460770396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.460770396\) |
| \(L(2)\) |
|
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 | \( 1 - 3.34iT - 25T^{2} \) |
| 7 | \( 1 - 5.01T + 49T^{2} \) |
| 11 | \( 1 + 7.85iT - 121T^{2} \) |
| 13 | \( 1 + 14.8T + 169T^{2} \) |
| 17 | \( 1 + 21.2iT - 289T^{2} \) |
| 19 | \( 1 - 24.0T + 361T^{2} \) |
| 23 | \( 1 - 21.0iT - 529T^{2} \) |
| 29 | \( 1 + 31.4iT - 841T^{2} \) |
| 31 | \( 1 - 4.08T + 961T^{2} \) |
| 37 | \( 1 + 41.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 4.27iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 16.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 66.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 63.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 76.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 99.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 88.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 5.01iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 38.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 38.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 142. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 86.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 164.T + 9.40e3T^{2} \) |
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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
−8.240419006240244732228088203889, −7.55278155301113140966903944525, −7.08749441979131807235194546988, −6.10818100098031861782912645848, −5.17869409565090592533280703618, −4.70228150170795003530519396578, −3.33931264696133612062968388135, −2.81623006575409274607514182575, −1.63983144838151038649968898876, −0.34070036601672792771542000066,
1.15328643194635843583889813537, 1.97721176583220737938116944721, 3.11013092209671437495434891972, 4.26850555348178623488172585669, 4.93736813149211167075583276132, 5.43263111980316830170501316145, 6.61828806950227043717773833625, 7.35235041611702094933148511271, 8.018915286428604692305861156053, 8.797150029170441980412552817729
