L(s) = 1 | − 2.82i·2-s + 16.1·3-s − 8.00·4-s + 27.1·5-s − 45.8i·6-s − 45.4i·7-s + 22.6i·8-s + 181.·9-s − 76.7i·10-s − 129.·12-s − 156. i·13-s − 128.·14-s + 439.·15-s + 64.0·16-s + 326. i·17-s − 512. i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.79·3-s − 0.500·4-s + 1.08·5-s − 1.27i·6-s − 0.926i·7-s + 0.353i·8-s + 2.23·9-s − 0.767i·10-s − 0.899·12-s − 0.926i·13-s − 0.655·14-s + 1.95·15-s + 0.250·16-s + 1.12i·17-s − 1.58i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(4.191736879\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.191736879\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82iT \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 16.1T + 81T^{2} \) |
| 5 | \( 1 - 27.1T + 625T^{2} \) |
| 7 | \( 1 + 45.4iT - 2.40e3T^{2} \) |
| 13 | \( 1 + 156. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 326. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 106. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 467.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 177. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 272.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.13e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.29e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.33e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.63e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.63e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 4.40e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 4.75e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 1.18e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.31e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 1.89e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 1.27e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 2.31e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.38e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.90e3T + 8.85e7T^{2} \) |
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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
−10.91146817383559372368635771095, −9.917141481400510203051830168805, −9.649421696056158526401291037947, −8.352260246115375980201577948834, −7.74311547218557457716231864191, −6.20195166331843931726148115473, −4.45309643462725223670437623014, −3.40320712934619148805947540234, −2.34915863374754442673153112468, −1.28591630290826622197158312845,
1.83526131280092844134619641031, 2.74283855799204319646661799562, 4.22441119768393540445238585609, 5.61817886569236264499447394943, 6.79694332890974858158933465229, 7.86432409179790659482750352656, 8.861632359232485292747677037925, 9.359217672894512709559799797114, 10.05547342933799838030399045465, 11.93601875295268118613280768482