| L(s) = 1 | − 2-s + 4-s − 3.35i·5-s + i·7-s − 8-s + 3.35i·10-s + (3.06 + 1.25i)11-s + 2.62i·13-s − i·14-s + 16-s + 2.58·17-s + 7.21i·19-s − 3.35i·20-s + (−3.06 − 1.25i)22-s + 5.09i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.49i·5-s + 0.377i·7-s − 0.353·8-s + 1.06i·10-s + (0.925 + 0.379i)11-s + 0.728i·13-s − 0.267i·14-s + 0.250·16-s + 0.627·17-s + 1.65i·19-s − 0.749i·20-s + (−0.654 − 0.268i)22-s + 1.06i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.224000146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224000146\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.06 - 1.25i)T \) |
| good | 5 | \( 1 + 3.35iT - 5T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 - 7.21iT - 19T^{2} \) |
| 23 | \( 1 - 5.09iT - 23T^{2} \) |
| 29 | \( 1 + 1.96T + 29T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 9.07T + 41T^{2} \) |
| 43 | \( 1 - 4.02iT - 43T^{2} \) |
| 47 | \( 1 - 7.16iT - 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 11.5iT - 59T^{2} \) |
| 61 | \( 1 - 4.07iT - 61T^{2} \) |
| 67 | \( 1 + 2.06T + 67T^{2} \) |
| 71 | \( 1 - 11.7iT - 71T^{2} \) |
| 73 | \( 1 + 5.29iT - 73T^{2} \) |
| 79 | \( 1 + 3.51iT - 79T^{2} \) |
| 83 | \( 1 + 2.98T + 83T^{2} \) |
| 89 | \( 1 + 12.8iT - 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−9.423237868515051386717117204716, −9.004918011780703012578738943183, −8.052393870322646228315567039524, −7.51481196554280848452571060967, −6.21720838156522788843642247701, −5.59722674554417633686193235410, −4.49582266447222541775528216990, −3.62575850215829828100666839254, −1.89946809741231699795797621598, −1.16086909669072413488558159279,
0.75599138910731407523182643747, 2.44640888760932329103896920410, 3.16996588493837766670764824083, 4.22694436630111971361796267084, 5.73056969410415638254294872312, 6.49531119765329362567375299333, 7.18463308467619712263358013658, 7.76166798653622697158115419138, 8.868056526707064200617294164266, 9.546272901664433233186389984405
