| L(s) = 1 | + 2.47i·2-s − 2.12·4-s + 2.55·5-s + 0.170i·7-s + 4.63i·8-s + 6.32i·10-s − 19.2i·13-s − 0.421·14-s − 19.9·16-s + 20.2i·17-s + 14.5i·19-s − 5.43·20-s − 7.74·23-s − 18.4·25-s + 47.7·26-s + ⋯ |
| L(s) = 1 | + 1.23i·2-s − 0.532·4-s + 0.510·5-s + 0.0243i·7-s + 0.579i·8-s + 0.632i·10-s − 1.48i·13-s − 0.0301·14-s − 1.24·16-s + 1.19i·17-s + 0.767i·19-s − 0.271·20-s − 0.336·23-s − 0.739·25-s + 1.83·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.517098323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.517098323\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.47iT - 4T^{2} \) |
| 5 | \( 1 - 2.55T + 25T^{2} \) |
| 7 | \( 1 - 0.170iT - 49T^{2} \) |
| 13 | \( 1 + 19.2iT - 169T^{2} \) |
| 17 | \( 1 - 20.2iT - 289T^{2} \) |
| 19 | \( 1 - 14.5iT - 361T^{2} \) |
| 23 | \( 1 + 7.74T + 529T^{2} \) |
| 29 | \( 1 - 38.2iT - 841T^{2} \) |
| 31 | \( 1 + 42.3T + 961T^{2} \) |
| 37 | \( 1 - 51.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 46.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 59.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 34.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 15.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 26.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 63.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 2.91T + 4.48e3T^{2} \) |
| 71 | \( 1 + 96.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 17.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 52.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 23.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 97.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 50.6T + 9.40e3T^{2} \) |
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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
−10.05237028023244322999280703774, −9.069815907427081304864093091696, −8.106272723040122201650791095129, −7.76486670542077100818113501248, −6.67890229546303901702591686631, −5.76779378039870204400136896919, −5.54568499374066060117923069541, −4.20673689491032507378571982322, −2.92206385031908631744452977190, −1.57897356956707809081001798807,
0.43259005494844178760100555540, 1.88899772931897919518561366811, 2.47879792501923357330705763142, 3.76640822086431111894409085666, 4.53586018557819789540428078717, 5.74324695221507512702031047083, 6.77355847330847746178086654539, 7.45514327193190728043787531259, 8.925987951609192582633520508349, 9.427737546644470996436867922078
