| L(s) = 1 | + (0.841 + 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (0.415 − 0.909i)6-s + (−1.70 + 1.96i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (−0.654 − 0.755i)10-s + (−4.94 + 3.17i)11-s + (0.841 − 0.540i)12-s + (−2.86 − 3.31i)13-s + (−2.49 + 0.732i)14-s + (−0.142 + 0.989i)15-s + (−0.654 + 0.755i)16-s + (−1.64 + 3.61i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.0821 − 0.571i)3-s + (0.207 + 0.454i)4-s + (−0.429 − 0.125i)5-s + (0.169 − 0.371i)6-s + (−0.643 + 0.742i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (−0.207 − 0.238i)10-s + (−1.49 + 0.957i)11-s + (0.242 − 0.156i)12-s + (−0.795 − 0.918i)13-s + (−0.666 + 0.195i)14-s + (−0.0367 + 0.255i)15-s + (−0.163 + 0.188i)16-s + (−0.399 + 0.875i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0992204 + 0.552450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0992204 + 0.552450i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (-2.72 - 3.94i)T \) |
| good | 7 | \( 1 + (1.70 - 1.96i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (4.94 - 3.17i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (2.86 + 3.31i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.64 - 3.61i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.36 + 2.98i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.46 + 7.59i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (1.22 - 8.53i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (4.36 - 1.28i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.08 - 0.612i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.836 + 5.82i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 0.734T + 47T^{2} \) |
| 53 | \( 1 + (9.14 - 10.5i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.78 - 3.21i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.35 + 9.40i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (5.40 + 3.47i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-9.10 - 5.85i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.63 + 5.76i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.290 - 0.334i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-1.80 + 0.529i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.121 - 0.842i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.82 + 1.12i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.90694178229874762571619792236, −10.08717726722610904396458481929, −8.916574037651669489736271301802, −7.967263425506447696818135042778, −7.36168074270455004314641253217, −6.41635688256054686275058523095, −5.39727146712941637632142818693, −4.72380358707781283935933808578, −3.15039229938657112016381370641, −2.30998331782139411599376212783,
0.22351614666659382615737559405, 2.57117213459469418629621915702, 3.45437334195813108743269406588, 4.49113935503365680597085100088, 5.26346014699205817681872522765, 6.46630640211602884087702091977, 7.30760822220676780575748333965, 8.409734088807781820325829949203, 9.502841790957569419026569286139, 10.30092215775376370048938335260
