| 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 + (−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 + (2.37 + 0.866i)15-s + (−2.75 − 2.31i)16-s + (−0.0812 + 0.460i)17-s − 7.12·18-s + (4.29 − 0.725i)19-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 + (−0.520 + 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 + (0.614 + 0.223i)15-s + (−0.688 − 0.577i)16-s + (−0.0197 + 0.111i)17-s − 1.68·18-s + (0.986 − 0.166i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.692545 + 2.46981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.692545 + 2.46981i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 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 | \( 1 + (-0.826 + 0.300i)T + (3.83 - 3.21i)T^{2} \) |
| 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 | \( 1 + (1.64 - 9.30i)T + (-91.1 - 33.1i)T^{2} \) |
| 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
−9.898614888022525502122017883317, −9.416070277876504943581919446743, −8.776495330334467763141442783680, −7.909474499594316829901737609853, −7.31832040785452583865429577815, −6.09538047137273653543621349099, −5.25263991921619110262957692062, −4.15218143297759120521883913393, −3.12561359403715101987065163012, −2.15883977800196062590632531842,
1.17423160387027861533115835375, 2.08733550620531473678310913992, 2.88195871190289616660914807431, 3.70420735777308371571942832606, 5.36446571330787161420530094993, 6.77028453442581161444392335763, 7.09644708662684280212681649096, 8.115366053933841749490187664222, 9.056996451997079330083234664827, 9.693658761276909701767831789230
