| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.427674 - 0.109301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.427674 - 0.109301i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (4.29 + 0.725i)T \) |
| 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
−18.61706599244867741133638981998, −17.19948501361039861101182927129, −16.07343761478541758197989390165, −15.11951850740594029295116528886, −12.53544018770985978205337199857, −11.49281066119051464488966881663, −10.63686474955047339161046226295, −9.394457177278772894493182185997, −6.33180420818185583642667808859, −4.23429636524392747738706203360,
5.79418956427122526140543409541, 6.84536631178454262481636256613, 8.154055427497165739338416140603, 11.01900410709222093023054735482, 11.95024484915269760046238268468, 13.42255803319803545560594769682, 15.23795684430034497437414822726, 16.53071355777576816660650010258, 17.28178374527330375218651793907, 18.33669529504490730858984858080
