| 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
−11.25621744677002143811617810719, −10.51180672545867127653260505397, −9.637788337442870960089274287604, −8.967470598147416002476062436102, −7.85189774556330197239105515972, −6.87510090720365768916465353254, −4.97387643397946197799632094972, −3.76520970717884104479172336968, −3.32515090172289908154518926700, −2.05658455018186948127697739483,
1.78437185774803154003265638588, 2.97095999762589176156766316467, 4.56187641016747336658110071447, 6.00893342448253763061383295768, 7.00170844568200416853318308032, 7.59067069200109754853769250691, 8.300253276659001934606668746718, 9.180991653862622995836312489466, 10.43186368145497267064423941195, 11.84580401871674299867461609804
