| L(s) = 1 | − 2.20i·2-s − 0.875·4-s − 6.11i·5-s + 0.579·7-s − 6.89i·8-s − 13.5·10-s + 7.56i·11-s − 1.13·13-s − 1.27i·14-s − 18.7·16-s − 16.3i·17-s − 34.7·19-s + 5.35i·20-s + 16.7·22-s − 21.2i·23-s + ⋯ |
| L(s) = 1 | − 1.10i·2-s − 0.218·4-s − 1.22i·5-s + 0.0827·7-s − 0.862i·8-s − 1.35·10-s + 0.687i·11-s − 0.0875·13-s − 0.0913i·14-s − 1.17·16-s − 0.959i·17-s − 1.82·19-s + 0.267i·20-s + 0.759·22-s − 0.922i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.346105665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.346105665\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 - 8.42iT \) |
| good | 2 | \( 1 + 2.20iT - 4T^{2} \) |
| 5 | \( 1 + 6.11iT - 25T^{2} \) |
| 7 | \( 1 - 0.579T + 49T^{2} \) |
| 11 | \( 1 - 7.56iT - 121T^{2} \) |
| 13 | \( 1 + 1.13T + 169T^{2} \) |
| 17 | \( 1 + 16.3iT - 289T^{2} \) |
| 19 | \( 1 + 34.7T + 361T^{2} \) |
| 23 | \( 1 + 21.2iT - 529T^{2} \) |
| 29 | \( 1 - 5.66iT - 841T^{2} \) |
| 31 | \( 1 - 23.8T + 961T^{2} \) |
| 37 | \( 1 - 11.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 2.27iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 21.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 18.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 0.902iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 59.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 82.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 9.45T + 4.48e3T^{2} \) |
| 73 | \( 1 + 85.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 14.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 33.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 81.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 43.4T + 9.40e3T^{2} \) |
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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
−9.990704376095905080499404086097, −9.130784040091408112623363679530, −8.432625619735710255606483586274, −7.21645841934549563274381052854, −6.21170527009241443935198189499, −4.74190666400523263403627673939, −4.28912730383416002222593113964, −2.75554647133978351068782429637, −1.72517114537179066991087281209, −0.46334250187852499984684881920,
2.09891069902352190004258715997, 3.27674105462611446086768142814, 4.58945012132787460500747083258, 6.05903125712338415938231635934, 6.26559623311871414074082226676, 7.27162217863441747532043449474, 8.069707835749350989360428584595, 8.805617874240358403774972230915, 10.13265924436106518438303034908, 10.93693258795747249214094287732
