| L(s) = 1 | − 2.73·3-s + 0.732i·7-s + 4.46·9-s − 2i·11-s − 3.46·13-s − 3.46i·17-s − 0.535i·19-s − 2i·21-s + 6.19i·23-s − 3.99·27-s + 6.92i·29-s + 5.46·31-s + 5.46i·33-s + 2·37-s + 9.46·39-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 0.276i·7-s + 1.48·9-s − 0.603i·11-s − 0.960·13-s − 0.840i·17-s − 0.122i·19-s − 0.436i·21-s + 1.29i·23-s − 0.769·27-s + 1.28i·29-s + 0.981·31-s + 0.951i·33-s + 0.328·37-s + 1.51·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.675286 + 0.256281i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.675286 + 0.256281i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 - 0.732iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 0.535iT - 19T^{2} \) |
| 23 | \( 1 - 6.19iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 - 3.26iT - 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 7.46iT - 59T^{2} \) |
| 61 | \( 1 - 8.92iT - 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 - 7.46iT - 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 - 14.3iT - 97T^{2} \) |
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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
−10.50237643867361511975846330068, −9.719872995702805122124607713075, −8.795379713762190197949664647790, −7.47335822409195784518781253336, −6.84747094849268895337798638807, −5.72449984129949467271886448814, −5.30024333190363153729156849412, −4.27685926004464258658960108921, −2.75233416247857991061685120281, −0.952451500326555797888790247143,
0.59390591848011075680905120225, 2.28452231336834423247504603742, 4.15069628866297945897825031989, 4.80267018004132880107884374695, 5.81343840169316777039913360281, 6.54918519637901190568328726645, 7.34553595293496244776974282858, 8.363868200100559857074515288777, 9.731938269464152289447297588505, 10.26930631948117354974895097254
