L(s) = 1 | + (−2.5 + 4.33i)2-s + (−8.50 − 14.7i)4-s + (2.5 + 4.33i)5-s + (15 − 25.9i)7-s + 45.0·8-s − 25.0·10-s + (−25 + 43.3i)11-s + (10 + 17.3i)13-s + (75.0 + 129. i)14-s + (−44.5 + 77.0i)16-s − 10·17-s − 44·19-s + (42.5 − 73.6i)20-s + (−125 − 216. i)22-s + (−60 − 103. i)23-s + ⋯ |
L(s) = 1 | + (−0.883 + 1.53i)2-s + (−1.06 − 1.84i)4-s + (0.223 + 0.387i)5-s + (0.809 − 1.40i)7-s + 1.98·8-s − 0.790·10-s + (−0.685 + 1.18i)11-s + (0.213 + 0.369i)13-s + (1.43 + 2.47i)14-s + (−0.695 + 1.20i)16-s − 0.142·17-s − 0.531·19-s + (0.475 − 0.823i)20-s + (−1.21 − 2.09i)22-s + (−0.543 − 0.942i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4719820540\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4719820540\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 2 | \( 1 + (2.5 - 4.33i)T + (-4 - 6.92i)T^{2} \) |
| 7 | \( 1 + (-15 + 25.9i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (25 - 43.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-10 - 17.3i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 10T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44T + 6.85e3T^{2} \) |
| 23 | \( 1 + (60 + 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-25 + 43.3i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (54 + 93.5i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 40T + 5.06e4T^{2} \) |
| 41 | \( 1 + (200 + 346. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (140 - 242. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-140 + 242. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 610T + 1.48e5T^{2} \) |
| 59 | \( 1 + (25 + 43.3i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-259 + 448. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-90 - 155. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 700T + 3.57e5T^{2} \) |
| 73 | \( 1 + 410T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-258 + 446. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (330 - 571. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-815 + 1.41e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.30666160217766735405054743280, −9.842876180679034293157990146802, −8.571891976392399784184266441378, −7.78274463408238423464839885345, −7.11891491713818793191777992537, −6.40565948983472398334174893231, −5.07638512658995306850145762071, −4.22877128469050909203900850297, −1.83196539973514570274446355534, −0.22064764322540520113011834597,
1.34940713884057245615358860871, 2.39407947475191098804945421754, 3.40066265727424851096887774839, 4.96900613364647962262309670031, 5.93029534567754306602593153570, 7.961726012138902725451146265600, 8.494686268336324330294094671012, 9.100788333211561553882095945943, 10.12289939200531715862184050419, 11.04271737016092636847719941447