| L(s) = 1 | + (0.990 + 0.139i)2-s + (−0.689 + 2.76i)3-s + (0.961 + 0.275i)4-s + (−0.144 + 4.15i)5-s + (−1.06 + 2.64i)6-s + (1.74 − 0.778i)7-s + (0.913 + 0.406i)8-s + (−4.52 − 2.40i)9-s + (−0.721 + 4.09i)10-s + (1.86 − 2.74i)11-s + (−1.42 + 2.46i)12-s + (1.69 − 1.05i)13-s + (1.84 − 0.527i)14-s + (−11.3 − 3.26i)15-s + (0.848 + 0.529i)16-s + (0.677 − 0.360i)17-s + ⋯ |
| L(s) = 1 | + (0.700 + 0.0984i)2-s + (−0.398 + 1.59i)3-s + (0.480 + 0.137i)4-s + (−0.0648 + 1.85i)5-s + (−0.435 + 1.07i)6-s + (0.661 − 0.294i)7-s + (0.322 + 0.143i)8-s + (−1.50 − 0.801i)9-s + (−0.228 + 1.29i)10-s + (0.562 − 0.826i)11-s + (−0.411 + 0.712i)12-s + (0.469 − 0.293i)13-s + (0.491 − 0.141i)14-s + (−2.93 − 0.842i)15-s + (0.212 + 0.132i)16-s + (0.164 − 0.0874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.731729 + 1.79344i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.731729 + 1.79344i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.990 - 0.139i)T \) |
| 11 | \( 1 + (-1.86 + 2.74i)T \) |
| 19 | \( 1 + (0.328 + 4.34i)T \) |
| good | 3 | \( 1 + (0.689 - 2.76i)T + (-2.64 - 1.40i)T^{2} \) |
| 5 | \( 1 + (0.144 - 4.15i)T + (-4.98 - 0.348i)T^{2} \) |
| 7 | \( 1 + (-1.74 + 0.778i)T + (4.68 - 5.20i)T^{2} \) |
| 13 | \( 1 + (-1.69 + 1.05i)T + (5.69 - 11.6i)T^{2} \) |
| 17 | \( 1 + (-0.677 + 0.360i)T + (9.50 - 14.0i)T^{2} \) |
| 23 | \( 1 + (2.10 - 0.764i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-4.56 + 4.40i)T + (1.01 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-4.67 - 5.18i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (7.37 + 5.35i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.920 + 3.69i)T + (-36.2 - 19.2i)T^{2} \) |
| 43 | \( 1 + (-1.75 - 0.638i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-4.12 - 8.46i)T + (-28.9 + 37.0i)T^{2} \) |
| 53 | \( 1 + (-0.337 - 9.66i)T + (-52.8 + 3.69i)T^{2} \) |
| 59 | \( 1 + (-2.38 + 4.88i)T + (-36.3 - 46.4i)T^{2} \) |
| 61 | \( 1 + (1.09 + 1.39i)T + (-14.7 + 59.1i)T^{2} \) |
| 67 | \( 1 + (-2.05 + 11.6i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.115 - 3.31i)T + (-70.8 - 4.95i)T^{2} \) |
| 73 | \( 1 + (15.9 - 1.11i)T + (72.2 - 10.1i)T^{2} \) |
| 79 | \( 1 + (4.41 + 10.9i)T + (-56.8 + 54.8i)T^{2} \) |
| 83 | \( 1 + (-13.0 - 2.76i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (5.37 + 4.51i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-5.02 - 0.706i)T + (93.2 + 26.7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.22370236663849317234923680075, −10.76557870982192283327427407749, −10.24003128603622091966677318076, −8.996438825524631549869793657350, −7.69031202262762139696267888467, −6.52193100657528419001573936513, −5.79701965112373612733695947954, −4.56398425818472916079205256170, −3.66175594640378098869325107445, −2.86596878496029275925642788911,
1.22944863849484719122206099748, 1.91609241071106582854206518437, 4.17427675756292729693255925408, 5.15296160254822089545873947374, 5.97264632772413633113514285291, 7.00318260143468431598812516732, 8.144864849948282240646830552557, 8.589893833150794354133806849504, 10.00693029699370427018019787163, 11.66787903714326545567091041468
