| L(s) = 1 | − 1.06i·2-s + 0.867·4-s − 3.23·5-s − 3.05i·8-s + 3.44i·10-s − 0.699i·11-s + 2.71i·13-s − 1.51·16-s + 6.50·17-s − 0.581i·19-s − 2.80·20-s − 0.744·22-s − 0.742i·23-s + 5.47·25-s + 2.89·26-s + ⋯ |
| L(s) = 1 | − 0.752i·2-s + 0.433·4-s − 1.44·5-s − 1.07i·8-s + 1.08i·10-s − 0.210i·11-s + 0.754i·13-s − 0.377·16-s + 1.57·17-s − 0.133i·19-s − 0.628·20-s − 0.158·22-s − 0.154i·23-s + 1.09·25-s + 0.567·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.448373114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.448373114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.06iT - 2T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 + 0.699iT - 11T^{2} \) |
| 13 | \( 1 - 2.71iT - 13T^{2} \) |
| 17 | \( 1 - 6.50T + 17T^{2} \) |
| 19 | \( 1 + 0.581iT - 19T^{2} \) |
| 23 | \( 1 + 0.742iT - 23T^{2} \) |
| 29 | \( 1 + 6.65iT - 29T^{2} \) |
| 31 | \( 1 - 8.87iT - 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 - 2.32T + 47T^{2} \) |
| 53 | \( 1 + 6.07iT - 53T^{2} \) |
| 59 | \( 1 - 0.355T + 59T^{2} \) |
| 61 | \( 1 + 11.7iT - 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 8.84iT - 71T^{2} \) |
| 73 | \( 1 + 14.2iT - 73T^{2} \) |
| 79 | \( 1 + 0.775T + 79T^{2} \) |
| 83 | \( 1 + 16.5T + 83T^{2} \) |
| 89 | \( 1 + 3.83T + 89T^{2} \) |
| 97 | \( 1 - 10.4iT - 97T^{2} \) |
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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
−8.269715645629554930224881891750, −7.78702234125892269492936643284, −7.00949426117335648382433400018, −6.36149433601974153413254628004, −5.20983837835203280148252309417, −4.20160191633382044862793172498, −3.56080480230450501494689937631, −2.90809230639854717527181015188, −1.66224125619691727880430950328, −0.52264593048983123857511235511,
1.08742005246492141049734473268, 2.63209534026759126139199295924, 3.45855822489214145624194960761, 4.30057598200103980016131580038, 5.38914523277494050636455293612, 5.85167966320722700101799567238, 7.06493714109911788174179245096, 7.38160575423468300269138371388, 8.106722587099959010306592003079, 8.453384052561917265593767802602
