L(s) = 1 | + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + (1.91 − 3.31i)5-s + (−2.09 − 1.62i)7-s + (−2 − 1.99i)8-s + (5.22 + 1.40i)10-s + (−3.82 + 2.20i)11-s − 5.41i·13-s + (1.44 − 3.44i)14-s + (1.99 − 3.46i)16-s + (1.94 − 1.12i)17-s + (2.70 − 4.68i)19-s + 7.65i·20-s + (−4.41 − 4.41i)22-s + (−1.70 + 2.95i)23-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + (0.856 − 1.48i)5-s + (−0.790 − 0.612i)7-s + (−0.707 − 0.707i)8-s + (1.65 + 0.443i)10-s + (−1.15 + 0.665i)11-s − 1.50i·13-s + (0.387 − 0.921i)14-s + (0.499 − 0.866i)16-s + (0.471 − 0.271i)17-s + (0.621 − 1.07i)19-s + 1.71i·20-s + (−0.941 − 0.941i)22-s + (−0.355 + 0.616i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.14523 - 0.461505i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14523 - 0.461505i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.09 + 1.62i)T \) |
good | 5 | \( 1 + (-1.91 + 3.31i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.82 - 2.20i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.41iT - 13T^{2} \) |
| 17 | \( 1 + (-1.94 + 1.12i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.70 + 4.68i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.70 - 2.95i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + (1.37 - 0.792i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.39 - 2.53i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.17iT - 41T^{2} \) |
| 43 | \( 1 - 8.58T + 43T^{2} \) |
| 47 | \( 1 + (-2.70 + 4.68i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.91 - 10.2i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.25 + 3.03i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.94 + 1.12i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.82 + 3.16i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.89T + 71T^{2} \) |
| 73 | \( 1 + (7.65 + 13.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.25 + 3.03i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.07iT - 83T^{2} \) |
| 89 | \( 1 + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−10.40716809024656483612299899282, −9.691660176200755478931760862630, −9.058585078471190170237199477802, −7.900502340525525552014932001233, −7.32515551603728179908518599576, −5.85428843686856879287947638716, −5.35220671302165710260540577736, −4.46470360014473993305892644788, −2.96283566005710961424221677802, −0.67004294413124188324539306847,
2.10829705496825386771136473893, 2.86607746189045199990371475019, 3.88072744281486813196806707693, 5.68037609748578231400032013035, 6.01172827869766397113197575287, 7.27551827615902118593275944680, 8.679874487096463650421052688398, 9.686722180159022035095301194801, 10.15826105996586740745770163295, 10.97458992769048703584048389349