L(s) = 1 | + (−2.05 − 3.43i)2-s + 15.5i·3-s + (−7.56 + 14.0i)4-s + 11.1·5-s + (53.4 − 31.9i)6-s + 37.6i·7-s + (63.9 − 2.96i)8-s − 161.·9-s + (−22.9 − 38.3i)10-s − 26.6i·11-s + (−219. − 117. i)12-s + 58.0·13-s + (129. − 77.2i)14-s + 174. i·15-s + (−141. − 213. i)16-s + 467.·17-s + ⋯ |
L(s) = 1 | + (−0.513 − 0.858i)2-s + 1.73i·3-s + (−0.473 + 0.881i)4-s + 0.447·5-s + (1.48 − 0.888i)6-s + 0.767i·7-s + (0.998 − 0.0462i)8-s − 1.99·9-s + (−0.229 − 0.383i)10-s − 0.220i·11-s + (−1.52 − 0.818i)12-s + 0.343·13-s + (0.658 − 0.394i)14-s + 0.774i·15-s + (−0.552 − 0.833i)16-s + 1.61·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 - 0.881i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.473 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.832946 + 0.498198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.832946 + 0.498198i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.05 + 3.43i)T \) |
| 5 | \( 1 - 11.1T \) |
good | 3 | \( 1 - 15.5iT - 81T^{2} \) |
| 7 | \( 1 - 37.6iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 26.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 58.0T + 2.85e4T^{2} \) |
| 17 | \( 1 - 467.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 428. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 360. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 964.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 417. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.79e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 469.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 27.7iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.53e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 276.T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.81e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.05e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.16e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.68e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.00e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 705. iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.62e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.15e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.30e4T + 8.85e7T^{2} \) |
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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
−17.71128471867471916579042144442, −16.56659720842344102860144713839, −15.48161930000001253522284897289, −13.94270822900395804580731601155, −11.99735332445735081295093600647, −10.68920802850046118809626356272, −9.658613660633761313207384048830, −8.652051372507505909245174491005, −5.16383608956449348785813032079, −3.23616120169063072411840900659,
1.21650811634170806680548546729, 5.92709740218944042590620142340, 7.18970592916696548478377162420, 8.268115878597598656524037078044, 10.25972452007220088929205565447, 12.33840581399422521849784393257, 13.69301969067342275827241252318, 14.40745335476681804497076761796, 16.55543158628798460437593188327, 17.45836553472469711312126721014