L(s) = 1 | − 0.193i·2-s − i·3-s + 1.96·4-s + (−1.48 − 1.67i)5-s − 0.193·6-s + i·7-s − 0.768i·8-s − 9-s + (−0.324 + 0.287i)10-s + 2·11-s − 1.96i·12-s + 1.35i·13-s + 0.193·14-s + (−1.67 + 1.48i)15-s + 3.77·16-s + 3.35i·17-s + ⋯ |
L(s) = 1 | − 0.137i·2-s − 0.577i·3-s + 0.981·4-s + (−0.662 − 0.749i)5-s − 0.0791·6-s + 0.377i·7-s − 0.271i·8-s − 0.333·9-s + (−0.102 + 0.0908i)10-s + 0.603·11-s − 0.566i·12-s + 0.374i·13-s + 0.0518·14-s + (−0.432 + 0.382i)15-s + 0.943·16-s + 0.812i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01753 - 0.458536i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01753 - 0.458536i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.48 + 1.67i)T \) |
| 7 | \( 1 - iT \) |
good | 2 | \( 1 + 0.193iT - 2T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 1.35iT - 13T^{2} \) |
| 17 | \( 1 - 3.35iT - 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 - 4.96iT - 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 - 4.57T + 31T^{2} \) |
| 37 | \( 1 - 0.775iT - 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 12.6iT - 43T^{2} \) |
| 47 | \( 1 + 9.92iT - 47T^{2} \) |
| 53 | \( 1 - 8.57iT - 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 + 9.92iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 9.35iT - 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 - 18.4iT - 97T^{2} \) |
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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
−13.36413621830796445891106378661, −12.33513129041615351075814787645, −11.78205617523218564929812759355, −10.76797034640784383297070855636, −9.128776383263763143259776274522, −8.051970160515102683253813684499, −6.93928565868959444837122682974, −5.74893100189629764222583720633, −3.83329502351719482017700569137, −1.82985493990099968121407889957,
2.81996533810922616835865143105, 4.25559555092394245803359709341, 6.15715531072572172642212975156, 7.14629385346476907324518962572, 8.269554578974505741263299090642, 9.886200586564330919785774353055, 10.94048600234398559991926340665, 11.45762514181586991195022458648, 12.70914338834609206486145666453, 14.42500593966397533566738443130