L(s) = 1 | + (−1.5 − 2.59i)2-s + (−0.5 + 0.866i)4-s + (1.5 + 2.59i)5-s − 21·8-s + (4.5 − 7.79i)10-s + (−7.5 + 12.9i)11-s + 64·13-s + (35.5 + 61.4i)16-s + (−42 + 72.7i)17-s + (−8 − 13.8i)19-s − 3.00·20-s + 45·22-s + (−42 − 72.7i)23-s + (58 − 100. i)25-s + (−96 − 166. i)26-s + ⋯ |
L(s) = 1 | + (−0.530 − 0.918i)2-s + (−0.0625 + 0.108i)4-s + (0.134 + 0.232i)5-s − 0.928·8-s + (0.142 − 0.246i)10-s + (−0.205 + 0.356i)11-s + 1.36·13-s + (0.554 + 0.960i)16-s + (−0.599 + 1.03i)17-s + (−0.0965 − 0.167i)19-s − 0.0335·20-s + 0.436·22-s + (−0.380 − 0.659i)23-s + (0.464 − 0.803i)25-s + (−0.724 − 1.25i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.451656811\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.451656811\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.5 + 2.59i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-1.5 - 2.59i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (7.5 - 12.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 64T + 2.19e3T^{2} \) |
| 17 | \( 1 + (42 - 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (8 + 13.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (42 + 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 297T + 2.43e4T^{2} \) |
| 31 | \( 1 + (126.5 - 219. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-158 - 273. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 360T + 6.89e4T^{2} \) |
| 43 | \( 1 - 26T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-15 - 25.9i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-181.5 + 314. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-7.5 + 12.9i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (59 + 102. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-185 + 320. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 342T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-181 + 313. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (233.5 + 404. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 477T + 5.71e5T^{2} \) |
| 89 | \( 1 + (453 + 784. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 503T + 9.12e5T^{2} \) |
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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
−10.60982644287911418758961908882, −9.973218517059621998057799642773, −8.777504961730791300813302332479, −8.302807898883038704078072954677, −6.62527404408692767441820137638, −6.08138782445709783735751988771, −4.52550649009489103195520527079, −3.23648330000092306167548948425, −2.12193959654287018804306110627, −0.920477360918718739067526671656,
0.77496663956091385349843618612, 2.67373217283019854892413570213, 3.98766008989132867855067248652, 5.49575018011669787358588336015, 6.22841474261099443111631796615, 7.22791433762193334569516139643, 8.077111420380656843208190387772, 8.910746689049461258871995135307, 9.519499412395209552185180453650, 10.88592742798339387609587095604