L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.878 + 1.49i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (0.434 − 1.67i)6-s + 2.38·7-s + (0.707 − 0.707i)8-s + (−1.45 + 2.62i)9-s + 1.00·10-s − 5.76·11-s + (−1.49 + 0.878i)12-s + (−1.58 − 1.58i)13-s + (−1.68 − 1.68i)14-s + (−1.67 − 0.434i)15-s − 1.00·16-s + (0.766 − 0.766i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.507 + 0.861i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (0.177 − 0.684i)6-s + 0.901·7-s + (0.250 − 0.250i)8-s + (−0.485 + 0.874i)9-s + 0.316·10-s − 1.73·11-s + (−0.430 + 0.253i)12-s + (−0.438 − 0.438i)13-s + (−0.450 − 0.450i)14-s + (−0.432 − 0.112i)15-s − 0.250·16-s + (0.186 − 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7532162905\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7532162905\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.878 - 1.49i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 37 | \( 1 + (-5.79 - 1.85i)T \) |
good | 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 13 | \( 1 + (1.58 + 1.58i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.766 + 0.766i)T - 17iT^{2} \) |
| 19 | \( 1 + (-4.27 - 4.27i)T + 19iT^{2} \) |
| 23 | \( 1 + (5.87 - 5.87i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.666 - 0.666i)T + 29iT^{2} \) |
| 31 | \( 1 + (4.93 - 4.93i)T - 31iT^{2} \) |
| 41 | \( 1 + 2.85T + 41T^{2} \) |
| 43 | \( 1 + (6.47 + 6.47i)T + 43iT^{2} \) |
| 47 | \( 1 + 8.44iT - 47T^{2} \) |
| 53 | \( 1 - 7.21iT - 53T^{2} \) |
| 59 | \( 1 + (-5.63 + 5.63i)T - 59iT^{2} \) |
| 61 | \( 1 + (8.82 - 8.82i)T - 61iT^{2} \) |
| 67 | \( 1 - 2.96iT - 67T^{2} \) |
| 71 | \( 1 + 10.2iT - 71T^{2} \) |
| 73 | \( 1 - 1.48iT - 73T^{2} \) |
| 79 | \( 1 + (3.85 + 3.85i)T + 79iT^{2} \) |
| 83 | \( 1 - 16.2iT - 83T^{2} \) |
| 89 | \( 1 + (-3.88 - 3.88i)T + 89iT^{2} \) |
| 97 | \( 1 + (-11.4 - 11.4i)T + 97iT^{2} \) |
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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
−10.25115727543526653061429836720, −9.571151688775311793327007920298, −8.435107736574074398966749316088, −7.84325637196183736375624387644, −7.46939788975830235808397524467, −5.51769984633227275510893436812, −5.01944247915306777389984711492, −3.75469521542533265467642108122, −2.97345907348067394745903098814, −1.90218072171175919411337558829,
0.34507164394895710968578019081, 1.85995617732533763585448803659, 2.82878121688772365131567976172, 4.48113701250068291435025823076, 5.31241221887432088098444532163, 6.31233757781975323226309790403, 7.43245446838036018283738776334, 7.84955399024199043798953268464, 8.362335024966250875974922499951, 9.306560559158743708093189945159