| L(s) = 1 | − 2-s + 4-s + 2.50i·5-s + i·7-s − 8-s − 2.50i·10-s + (−3.23 − 0.715i)11-s − 1.06i·13-s − i·14-s + 16-s − 1.11·17-s − 0.185i·19-s + 2.50i·20-s + (3.23 + 0.715i)22-s + 5.70i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.11i·5-s + 0.377i·7-s − 0.353·8-s − 0.791i·10-s + (−0.976 − 0.215i)11-s − 0.296i·13-s − 0.267i·14-s + 0.250·16-s − 0.270·17-s − 0.0425i·19-s + 0.559i·20-s + (0.690 + 0.152i)22-s + 1.18i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1742209025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1742209025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (3.23 + 0.715i)T \) |
| good | 5 | \( 1 - 2.50iT - 5T^{2} \) |
| 13 | \( 1 + 1.06iT - 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 0.185iT - 19T^{2} \) |
| 23 | \( 1 - 5.70iT - 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 + 8.60T + 31T^{2} \) |
| 37 | \( 1 - 4.67T + 37T^{2} \) |
| 41 | \( 1 - 0.132T + 41T^{2} \) |
| 43 | \( 1 + 8.83iT - 43T^{2} \) |
| 47 | \( 1 + 5.08iT - 47T^{2} \) |
| 53 | \( 1 - 0.0980iT - 53T^{2} \) |
| 59 | \( 1 - 6.33iT - 59T^{2} \) |
| 61 | \( 1 + 3.93iT - 61T^{2} \) |
| 67 | \( 1 + 4.49T + 67T^{2} \) |
| 71 | \( 1 - 0.698iT - 71T^{2} \) |
| 73 | \( 1 - 0.169iT - 73T^{2} \) |
| 79 | \( 1 + 1.63iT - 79T^{2} \) |
| 83 | \( 1 + 6.27T + 83T^{2} \) |
| 89 | \( 1 + 15.2iT - 89T^{2} \) |
| 97 | \( 1 + 2.71T + 97T^{2} \) |
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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
−10.00932727523769630579950607824, −9.286808109740641497103036052269, −8.428659785838894465860141393182, −7.45805139443127681014908278407, −7.13141255293784722846336440174, −5.92097859970611580098424874547, −5.35913364510343049506380949256, −3.72136233350853812837523980902, −2.85759073734662374983154507262, −1.91940914386013910976867477197,
0.086467887705881206913669040487, 1.44562203573810299023705137479, 2.61100244451239904469230568257, 4.03874552312201490751397814617, 4.92213028116201988657175263610, 5.78230665930210276672634998196, 6.86282097918923166461480325046, 7.73103368917048333613371196224, 8.315176197954797190141215684630, 9.203535183829218849379843228886
