L(s) = 1 | − 1.93i·2-s − 1.73·4-s + (−1 + 2.44i)7-s − 0.517i·8-s + 5.27i·11-s − 6.69i·13-s + (4.73 + 1.93i)14-s − 4.46·16-s − 3.46·17-s − 6.69i·19-s + 10.1·22-s + 1.41i·23-s − 12.9·26-s + (1.73 − 4.24i)28-s − 1.41i·29-s + ⋯ |
L(s) = 1 | − 1.36i·2-s − 0.866·4-s + (−0.377 + 0.925i)7-s − 0.183i·8-s + 1.59i·11-s − 1.85i·13-s + (1.26 + 0.516i)14-s − 1.11·16-s − 0.840·17-s − 1.53i·19-s + 2.17·22-s + 0.294i·23-s − 2.53·26-s + (0.327 − 0.801i)28-s − 0.262i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4734771314\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4734771314\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
good | 2 | \( 1 + 1.93iT - 2T^{2} \) |
| 11 | \( 1 - 5.27iT - 11T^{2} \) |
| 13 | \( 1 + 6.69iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6.69iT - 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 1.79iT - 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 - 4.52iT - 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 + 3.58iT - 61T^{2} \) |
| 67 | \( 1 + 2.92T + 67T^{2} \) |
| 71 | \( 1 + 9.41iT - 71T^{2} \) |
| 73 | \( 1 - 6.69iT - 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 + 10.2iT - 97T^{2} \) |
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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
−9.201927693424201935724673481350, −8.410965395416398106653598114942, −7.25892733723252795743822814651, −6.53513597041654909270423705602, −5.27288602014921514800326649622, −4.61617643212068070660401879610, −3.35805011163688987684933260444, −2.64007372771614772705347750599, −1.82296152483570410160246725543, −0.17114046287340713083768345047,
1.73492033069647161129645230219, 3.40572981782655680787615359705, 4.23574759849997614264635815688, 5.22155650966124867975715477809, 6.25842290666714857585715401016, 6.63579756358362180508736033291, 7.33932506714161068177982705374, 8.449522191807926573076185829626, 8.677009781068355670986560715271, 9.748138296944242623515507445543