| L(s) = 1 | + (0.233 + 1.32i)2-s + (2.20 − 1.85i)3-s + (0.173 − 0.0632i)4-s + (−0.826 − 0.300i)5-s + (2.97 + 2.49i)6-s + (−0.173 + 0.300i)7-s + (1.47 + 2.54i)8-s + (0.918 − 5.21i)9-s + (0.205 − 1.16i)10-s + (1.11 + 1.92i)11-s + (0.266 − 0.460i)12-s + (−1.97 − 1.65i)13-s + (−0.439 − 0.160i)14-s + (−2.37 + 0.866i)15-s + (−2.75 + 2.31i)16-s + (0.0812 + 0.460i)17-s + ⋯ |
| L(s) = 1 | + (0.165 + 0.938i)2-s + (1.27 − 1.06i)3-s + (0.0868 − 0.0316i)4-s + (−0.369 − 0.134i)5-s + (1.21 + 1.01i)6-s + (−0.0656 + 0.113i)7-s + (0.520 + 0.901i)8-s + (0.306 − 1.73i)9-s + (0.0650 − 0.368i)10-s + (0.335 + 0.581i)11-s + (0.0768 − 0.133i)12-s + (−0.546 − 0.458i)13-s + (−0.117 − 0.0427i)14-s + (−0.614 + 0.223i)15-s + (−0.688 + 0.577i)16-s + (0.0197 + 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.23976 + 0.222124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23976 + 0.222124i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-0.233 - 1.32i)T + (-1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (-2.20 + 1.85i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (0.826 + 0.300i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.173 - 0.300i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.11 - 1.92i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.97 + 1.65i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0812 - 0.460i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.53 + 0.921i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.19 + 6.77i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (3.55 - 6.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 + (1.89 - 1.59i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.66 + 1.33i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.26 - 7.18i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (2.66 - 0.970i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.09 - 6.20i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (8.57 - 3.12i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.33 - 7.55i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (8.74 + 3.18i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.06 + 0.892i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-9.07 + 7.61i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.41 + 12.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.88 - 6.61i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (1.64 + 9.30i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.84580401871674299867461609804, −10.43186368145497267064423941195, −9.180991653862622995836312489466, −8.300253276659001934606668746718, −7.59067069200109754853769250691, −7.00170844568200416853318308032, −6.00893342448253763061383295768, −4.56187641016747336658110071447, −2.97095999762589176156766316467, −1.78437185774803154003265638588,
2.05658455018186948127697739483, 3.32515090172289908154518926700, 3.76520970717884104479172336968, 4.97387643397946197799632094972, 6.87510090720365768916465353254, 7.85189774556330197239105515972, 8.967470598147416002476062436102, 9.637788337442870960089274287604, 10.51180672545867127653260505397, 11.25621744677002143811617810719
