| L(s) = 1 | + i·2-s − 4-s + 2.97·5-s + (2.55 − 0.698i)7-s − i·8-s + 2.97i·10-s + i·11-s + 1.97i·13-s + (0.698 + 2.55i)14-s + 16-s − 1.57·17-s + 7.18i·19-s − 2.97·20-s − 22-s + 3.86·25-s − 1.97·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.33·5-s + (0.964 − 0.264i)7-s − 0.353i·8-s + 0.941i·10-s + 0.301i·11-s + 0.548i·13-s + (0.186 + 0.681i)14-s + 0.250·16-s − 0.382·17-s + 1.64i·19-s − 0.665·20-s − 0.213·22-s + 0.772·25-s − 0.387·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.247522672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.247522672\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.55 + 0.698i)T \) |
| 11 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.97T + 5T^{2} \) |
| 13 | \( 1 - 1.97iT - 13T^{2} \) |
| 17 | \( 1 + 1.57T + 17T^{2} \) |
| 19 | \( 1 - 7.18iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 1.30iT - 29T^{2} \) |
| 31 | \( 1 + 5.78iT - 31T^{2} \) |
| 37 | \( 1 - 8.04T + 37T^{2} \) |
| 41 | \( 1 - 3.81T + 41T^{2} \) |
| 43 | \( 1 - 9.14T + 43T^{2} \) |
| 47 | \( 1 + 2.63T + 47T^{2} \) |
| 53 | \( 1 - 8.94iT - 53T^{2} \) |
| 59 | \( 1 + 9.90T + 59T^{2} \) |
| 61 | \( 1 - 4.77iT - 61T^{2} \) |
| 67 | \( 1 - 0.699T + 67T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 + 8.20T + 89T^{2} \) |
| 97 | \( 1 + 8.41iT - 97T^{2} \) |
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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
−9.582325697823609275898335947039, −9.002966249786291446534969915169, −7.936358533833830483873468057526, −7.42787778545765348096074039558, −6.12313208609795313089977470552, −5.91615039406950092957215888393, −4.76445658216806958287806612219, −4.05207756501806328506517862233, −2.34961118969700263908668479900, −1.40705272634629840566816131325,
1.06127850023370383166992168535, 2.19991909662191238484008084676, 2.90150928370435566598616832204, 4.40845208035693087143323128105, 5.18627401720170343887647182287, 5.87515409237236702836937906184, 6.91623275051894421177675954794, 8.042024842309756515413150906195, 8.874803819417481125350259474043, 9.414637548528334718232350501375
