L(s) = 1 | + (1.72 + 0.158i)3-s + (1 + 1.73i)4-s + (−1.22 − 0.707i)5-s + (−1.32 + 2.29i)7-s + (2.94 + 0.548i)9-s + (2.23 + 2.45i)11-s + (1.44 + 3.14i)12-s − 4.58i·13-s + (−1.99 − 1.41i)15-s + (−1.99 + 3.46i)16-s + (−3.24 − 5.61i)17-s + (3.96 + 2.29i)19-s − 2.82i·20-s + (−2.64 + 3.74i)21-s + (−4.89 − 2.82i)23-s + ⋯ |
L(s) = 1 | + (0.995 + 0.0917i)3-s + (0.5 + 0.866i)4-s + (−0.547 − 0.316i)5-s + (−0.499 + 0.866i)7-s + (0.983 + 0.182i)9-s + (0.673 + 0.739i)11-s + (0.418 + 0.908i)12-s − 1.27i·13-s + (−0.516 − 0.365i)15-s + (−0.499 + 0.866i)16-s + (−0.785 − 1.36i)17-s + (0.910 + 0.525i)19-s − 0.632i·20-s + (−0.577 + 0.816i)21-s + (−1.02 − 0.589i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55159 + 0.575135i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55159 + 0.575135i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.72 - 0.158i)T \) |
| 7 | \( 1 + (1.32 - 2.29i)T \) |
| 11 | \( 1 + (-2.23 - 2.45i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.22 + 0.707i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 4.58iT - 13T^{2} \) |
| 17 | \( 1 + (3.24 + 5.61i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.96 - 2.29i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.89 + 2.82i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 - 4.58iT - 43T^{2} \) |
| 47 | \( 1 + (4.89 + 2.82i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.12 - 3.53i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.44 - 1.41i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.41iT - 71T^{2} \) |
| 73 | \( 1 + (-3.96 + 2.29i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (11.9 + 6.87i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.48T + 83T^{2} \) |
| 89 | \( 1 + (-13.4 - 7.77i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−12.27050104939260161334443015326, −11.76210515953642329206027496815, −10.16868435437713879786618549178, −9.217156323377959260443644899416, −8.285791987256648764268235095560, −7.59271364711123039560837686822, −6.47407100445051092900400334512, −4.64633702384744593101352024007, −3.39780205664226671070258017977, −2.42486330398899396924400138654,
1.59142481092930543791477979052, 3.33302812383531543684901904519, 4.34359762571092608234549159403, 6.37575360435973054009089114657, 6.92253265224252011914876726401, 8.104202916985935309579312177305, 9.292230307502014444977307735135, 10.05846915733883358684660070878, 11.10891220603883805716565898943, 11.89502392883359257761372290816