| L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.580 − 1.27i)3-s + (−0.959 + 0.281i)4-s + (−1.66 + 1.92i)5-s + (−1.34 − 0.393i)6-s + (1.75 − 1.12i)7-s + (0.415 + 0.909i)8-s + (0.684 + 0.790i)9-s + (2.14 + 1.37i)10-s + (−0.543 + 3.77i)11-s + (−0.198 + 1.38i)12-s + (−5.21 − 3.34i)13-s + (−1.36 − 1.57i)14-s + (1.47 + 3.23i)15-s + (0.841 − 0.540i)16-s + (−1.24 − 0.366i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.335 − 0.734i)3-s + (−0.479 + 0.140i)4-s + (−0.745 + 0.860i)5-s + (−0.547 − 0.160i)6-s + (0.663 − 0.426i)7-s + (0.146 + 0.321i)8-s + (0.228 + 0.263i)9-s + (0.677 + 0.435i)10-s + (−0.163 + 1.13i)11-s + (−0.0574 + 0.399i)12-s + (−1.44 − 0.929i)13-s + (−0.365 − 0.421i)14-s + (0.381 + 0.835i)15-s + (0.210 − 0.135i)16-s + (−0.303 − 0.0890i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.668509 - 0.382826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.668509 - 0.382826i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (1.33 + 4.60i)T \) |
| good | 3 | \( 1 + (-0.580 + 1.27i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (1.66 - 1.92i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-1.75 + 1.12i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.543 - 3.77i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (5.21 + 3.34i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.24 + 0.366i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-2.37 + 0.698i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (6.87 + 2.01i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.67 + 3.65i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-7.48 - 8.63i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.81 + 3.24i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (1.15 - 2.52i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 9.34T + 47T^{2} \) |
| 53 | \( 1 + (1.99 - 1.27i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.514 - 0.330i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.67 - 3.67i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (1.82 + 12.6i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.940 - 6.54i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.80 + 0.824i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (1.58 + 1.02i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (6.83 + 7.89i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.08 + 4.55i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (8.21 - 9.48i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−15.23793817603968364341350811752, −14.46922706635351300059453396917, −13.12440588978270356086932604608, −12.15435048058490674856239774954, −10.94999868662902427227811004453, −9.873421459610842650088191849862, −7.77864993695923013683794438563, −7.35304497069647927033949670993, −4.57577390071581444356933083800, −2.48180479336225520740870720323,
4.08648914218362542358378138902, 5.33374133263460441930161068605, 7.46955517919775576286674280241, 8.708304233574818071068039011735, 9.530305067490471344754983000681, 11.36004832073772459032465251426, 12.54079964726810714206406789030, 14.15264585062825555728467398378, 15.03806662565438958977523987330, 16.04535673610070360060680294749
