| L(s) = 1 | + (3.23 + 2.35i)2-s + (−0.559 − 5.32i)3-s + (4.94 + 15.2i)4-s + (23.3 + 40.3i)5-s + (10.7 − 18.5i)6-s + (1.01 + 0.215i)7-s + (−19.7 + 60.8i)8-s + (209. − 44.5i)9-s + (−19.5 + 185. i)10-s + (−441. + 490. i)11-s + (78.2 − 34.8i)12-s + (522. + 232. i)13-s + (2.77 + 3.07i)14-s + (201. − 146. i)15-s + (−207. + 150. i)16-s + (1.09e3 + 1.22e3i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.0358 − 0.341i)3-s + (0.154 + 0.475i)4-s + (0.417 + 0.722i)5-s + (0.121 − 0.210i)6-s + (0.00781 + 0.00166i)7-s + (−0.109 + 0.336i)8-s + (0.862 − 0.183i)9-s + (−0.0616 + 0.586i)10-s + (−1.10 + 1.22i)11-s + (0.156 − 0.0698i)12-s + (0.858 + 0.382i)13-s + (0.00378 + 0.00419i)14-s + (0.231 − 0.168i)15-s + (−0.202 + 0.146i)16-s + (0.922 + 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.96625 + 1.50693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96625 + 1.50693i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 31 | \( 1 + (-834. - 5.28e3i)T \) |
| good | 3 | \( 1 + (0.559 + 5.32i)T + (-237. + 50.5i)T^{2} \) |
| 5 | \( 1 + (-23.3 - 40.3i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-1.01 - 0.215i)T + (1.53e4 + 6.83e3i)T^{2} \) |
| 11 | \( 1 + (441. - 490. i)T + (-1.68e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-522. - 232. i)T + (2.48e5 + 2.75e5i)T^{2} \) |
| 17 | \( 1 + (-1.09e3 - 1.22e3i)T + (-1.48e5 + 1.41e6i)T^{2} \) |
| 19 | \( 1 + (-202. + 90.2i)T + (1.65e6 - 1.84e6i)T^{2} \) |
| 23 | \( 1 + (342. - 1.05e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (6.44e3 + 4.68e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 37 | \( 1 + (-6.86e3 + 1.18e4i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-1.72e3 + 1.63e4i)T + (-1.13e8 - 2.40e7i)T^{2} \) |
| 43 | \( 1 + (-8.81e3 + 3.92e3i)T + (9.83e7 - 1.09e8i)T^{2} \) |
| 47 | \( 1 + (5.49e3 - 3.99e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.39e3 + 508. i)T + (3.82e8 - 1.70e8i)T^{2} \) |
| 59 | \( 1 + (2.81e3 + 2.68e4i)T + (-6.99e8 + 1.48e8i)T^{2} \) |
| 61 | \( 1 - 3.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (5.34e3 + 9.25e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (3.02e4 - 6.42e3i)T + (1.64e9 - 7.33e8i)T^{2} \) |
| 73 | \( 1 + (2.34e4 - 2.60e4i)T + (-2.16e8 - 2.06e9i)T^{2} \) |
| 79 | \( 1 + (-6.29e4 - 6.98e4i)T + (-3.21e8 + 3.06e9i)T^{2} \) |
| 83 | \( 1 + (4.70e3 - 4.47e4i)T + (-3.85e9 - 8.18e8i)T^{2} \) |
| 89 | \( 1 + (-2.41e4 - 7.42e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-65.8 - 202. i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−14.25801314195564255449562910460, −13.10676649794313201138859180335, −12.41175971814115692093155227350, −10.81833280151650313198830770332, −9.754905787873528450420260146413, −7.85130897566127763166475117549, −6.90476016648538820613460764234, −5.66101989610622487670935231071, −3.93796332734738394920759037521, −2.03892100518710582191683633311,
1.09072130901234706518823926311, 3.18478069878952804817104229585, 4.86097083241518123218217516293, 5.85406816090451424593197675051, 7.84302810245935044695856993167, 9.338594316337804542860579554554, 10.44180725678364888564086974604, 11.47714392416369875543002659174, 13.13463148416987854670032090017, 13.25625551791816450454466033995
