L(s) = 1 | + (0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + 7i·7-s + (−4 − 6.92i)9-s + (−14.7 − 8.5i)11-s − 24·13-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (6.06 − 3.5i)19-s + (−3.5 + 6.06i)21-s + (6.06 − 3.5i)23-s + (12 − 20.7i)25-s − 17i·27-s − 24·29-s + (35.5 + 20.5i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.166i)3-s + (0.100 + 0.173i)5-s + i·7-s + (−0.444 − 0.769i)9-s + (−1.33 − 0.772i)11-s − 1.84·13-s + 0.0666i·15-s + (−0.0294 + 0.0509i)17-s + (0.319 − 0.184i)19-s + (−0.166 + 0.288i)21-s + (0.263 − 0.152i)23-s + (0.479 − 0.831i)25-s − 0.629i·27-s − 0.827·29-s + (1.14 + 0.661i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1843515320\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1843515320\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (14.7 + 8.5i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 24T + 169T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-6.06 + 3.5i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-6.06 + 3.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 24T + 841T^{2} \) |
| 31 | \( 1 + (-35.5 - 20.5i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (24.5 + 42.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 48T + 1.68e3T^{2} \) |
| 43 | \( 1 - 24iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (47.6 - 27.5i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (12.5 - 21.6i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-14.7 - 8.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-56.2 - 32.5i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 96iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (47.5 - 82.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (35.5 - 20.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 72iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (47.5 + 82.2i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 144T + 9.40e3T^{2} \) |
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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
−10.36993619783146990051045197061, −9.586956248161989057344243953538, −8.707375864934206021625182393542, −7.919854473579603577832382696820, −6.73740712523444394382001175166, −5.62491079840176051253075442472, −4.85657467050835926375502413978, −3.09322934258198497851951944187, −2.48510538827698115864032007646, −0.06667605047837658086524589844,
1.97773768532242015162113459173, 3.09491007957084283877198376297, 4.78685603516214480234950170554, 5.21411894544555510434594442273, 7.01427346443869412904306104481, 7.54949489544941922417424132344, 8.326397731645163418798219129378, 9.786548939182096424755283664698, 10.15506427610640399083195906953, 11.18610155457664639166436839888