| L(s) = 1 | + (−2.23 + 2i)3-s − 6i·7-s + (1.00 − 8.94i)9-s + 4.47i·11-s + 16i·13-s − 4.47·17-s − 2·19-s + (12 + 13.4i)21-s − 13.4·23-s + (15.6 + 22.0i)27-s − 31.3i·29-s + 18·31-s + (−8.94 − 10.0i)33-s + 16i·37-s + (−32 − 35.7i)39-s + ⋯ |
| L(s) = 1 | + (−0.745 + 0.666i)3-s − 0.857i·7-s + (0.111 − 0.993i)9-s + 0.406i·11-s + 1.23i·13-s − 0.263·17-s − 0.105·19-s + (0.571 + 0.638i)21-s − 0.583·23-s + (0.579 + 0.814i)27-s − 1.07i·29-s + 0.580·31-s + (−0.271 − 0.303i)33-s + 0.432i·37-s + (−0.820 − 0.917i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.169450458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169450458\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.23 - 2i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 6iT - 49T^{2} \) |
| 11 | \( 1 - 4.47iT - 121T^{2} \) |
| 13 | \( 1 - 16iT - 169T^{2} \) |
| 17 | \( 1 + 4.47T + 289T^{2} \) |
| 19 | \( 1 + 2T + 361T^{2} \) |
| 23 | \( 1 + 13.4T + 529T^{2} \) |
| 29 | \( 1 + 31.3iT - 841T^{2} \) |
| 31 | \( 1 - 18T + 961T^{2} \) |
| 37 | \( 1 - 16iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 62.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 16iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 49.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 4.47T + 2.80e3T^{2} \) |
| 59 | \( 1 + 4.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 82T + 3.72e3T^{2} \) |
| 67 | \( 1 - 24iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 138T + 6.24e3T^{2} \) |
| 83 | \( 1 - 93.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 166iT - 9.40e3T^{2} \) |
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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
−9.667480196246425468235207381055, −8.915370119984972737190847258092, −7.78248847157033286404702284423, −6.81214907909307161164452427003, −6.27689069787469570655043788782, −5.08917522522300372601671707190, −4.30055280063726088439808907998, −3.68701395721638939525838239366, −2.01999778584513188836074021663, −0.52006480552684253072183123665,
0.857819559536096679552470903875, 2.18846041561458016478118013203, 3.21992907080917550635149665124, 4.72487017414647091831688352258, 5.54958689365246866356610242361, 6.13949638893133785769813951083, 7.03440358749607194963855630911, 8.063125703336018754390213827108, 8.519077060367334186014283581256, 9.714662615205091977915897035150
