L(s) = 1 | − 1.08·3-s + 2.61i·5-s + (−2.61 − 0.414i)7-s − 1.82·9-s + 2i·11-s + 4.77i·13-s − 2.82i·15-s − 3.06i·17-s − 4.14·19-s + (2.82 + 0.448i)21-s + 7.65i·23-s − 1.82·25-s + 5.22·27-s + 3.65·29-s + 3.06·31-s + ⋯ |
L(s) = 1 | − 0.624·3-s + 1.16i·5-s + (−0.987 − 0.156i)7-s − 0.609·9-s + 0.603i·11-s + 1.32i·13-s − 0.730i·15-s − 0.742i·17-s − 0.950·19-s + (0.617 + 0.0978i)21-s + 1.59i·23-s − 0.365·25-s + 1.00·27-s + 0.679·29-s + 0.549·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.270294 + 0.530399i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.270294 + 0.530399i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.61 + 0.414i)T \) |
good | 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 - 2.61iT - 5T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 4.77iT - 13T^{2} \) |
| 17 | \( 1 + 3.06iT - 17T^{2} \) |
| 19 | \( 1 + 4.14T + 19T^{2} \) |
| 23 | \( 1 - 7.65iT - 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 + 9.55iT - 41T^{2} \) |
| 43 | \( 1 + 3.65iT - 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 8.47T + 59T^{2} \) |
| 61 | \( 1 + 2.61iT - 61T^{2} \) |
| 67 | \( 1 - 15.6iT - 67T^{2} \) |
| 71 | \( 1 + 8.82iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 12.8iT - 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 2.16iT - 89T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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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
−12.32684803815336509228777809815, −11.60416626693598911588003533582, −10.71489510247167660863075694386, −9.867598650526843706249521053776, −8.831847039028527280404642269121, −7.05280194098855487787461793762, −6.72775081693494630981319357845, −5.51196794458349198902874630592, −3.90825183869261432596270871156, −2.55417827599597349752748000297,
0.51298582969417518798000141339, 2.99836652766771273950472596065, 4.62584078268731707706650157106, 5.76019980429468459870069912118, 6.44448637825724201342894696109, 8.324715970035326596935536331890, 8.711717498378342289979042199049, 10.14151600116895064552821705362, 10.87130143194364847801986452375, 12.25652705913895321673477779145