| L(s) = 1 | + (−0.186 − 0.982i)2-s + (0.411 − 0.00964i)3-s + (−0.930 + 0.366i)4-s + (0.415 − 1.57i)5-s + (−0.0862 − 0.402i)6-s + (−0.208 − 0.464i)7-s + (0.533 + 0.845i)8-s + (−2.82 + 0.132i)9-s + (−1.62 − 0.114i)10-s + (−0.181 − 0.203i)11-s + (−0.379 + 0.159i)12-s + (0.295 − 4.19i)13-s + (−0.417 + 0.291i)14-s + (0.155 − 0.652i)15-s + (0.731 − 0.681i)16-s + (−2.44 − 2.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.131 − 0.694i)2-s + (0.237 − 0.00557i)3-s + (−0.465 + 0.183i)4-s + (0.185 − 0.704i)5-s + (−0.0351 − 0.164i)6-s + (−0.0787 − 0.175i)7-s + (0.188 + 0.299i)8-s + (−0.942 + 0.0442i)9-s + (−0.514 − 0.0362i)10-s + (−0.0546 − 0.0614i)11-s + (−0.109 + 0.0461i)12-s + (0.0819 − 1.16i)13-s + (−0.111 + 0.0778i)14-s + (0.0402 − 0.168i)15-s + (0.182 − 0.170i)16-s + (−0.593 − 0.527i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.217824 - 0.971037i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.217824 - 0.971037i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.186 + 0.982i)T \) |
| 269 | \( 1 + (14.9 - 6.77i)T \) |
| good | 3 | \( 1 + (-0.411 + 0.00964i)T + (2.99 - 0.140i)T^{2} \) |
| 5 | \( 1 + (-0.415 + 1.57i)T + (-4.34 - 2.46i)T^{2} \) |
| 7 | \( 1 + (0.208 + 0.464i)T + (-4.65 + 5.23i)T^{2} \) |
| 11 | \( 1 + (0.181 + 0.203i)T + (-1.28 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.295 + 4.19i)T + (-12.8 - 1.82i)T^{2} \) |
| 17 | \( 1 + (2.44 + 2.17i)T + (1.98 + 16.8i)T^{2} \) |
| 19 | \( 1 + (2.59 + 1.72i)T + (7.37 + 17.5i)T^{2} \) |
| 23 | \( 1 + (0.141 + 6.02i)T + (-22.9 + 1.07i)T^{2} \) |
| 29 | \( 1 + (-3.87 - 6.83i)T + (-14.8 + 24.8i)T^{2} \) |
| 31 | \( 1 + (4.64 + 0.547i)T + (30.1 + 7.20i)T^{2} \) |
| 37 | \( 1 + (-3.47 + 6.47i)T + (-20.4 - 30.8i)T^{2} \) |
| 41 | \( 1 + (2.28 + 0.434i)T + (38.1 + 15.0i)T^{2} \) |
| 43 | \( 1 + (3.06 - 1.74i)T + (22.0 - 36.8i)T^{2} \) |
| 47 | \( 1 + (3.54 + 1.02i)T + (39.7 + 25.0i)T^{2} \) |
| 53 | \( 1 + (2.96 + 4.04i)T + (-15.9 + 50.5i)T^{2} \) |
| 59 | \( 1 + (-4.59 - 11.6i)T + (-43.1 + 40.2i)T^{2} \) |
| 61 | \( 1 + (6.85 + 8.47i)T + (-12.7 + 59.6i)T^{2} \) |
| 67 | \( 1 + (-6.35 - 2.50i)T + (49.0 + 45.6i)T^{2} \) |
| 71 | \( 1 + (-3.21 + 4.60i)T + (-24.4 - 66.6i)T^{2} \) |
| 73 | \( 1 + (-0.0380 + 0.103i)T + (-55.6 - 47.2i)T^{2} \) |
| 79 | \( 1 + (-11.4 - 8.83i)T + (20.1 + 76.3i)T^{2} \) |
| 83 | \( 1 + (-0.934 - 9.94i)T + (-81.5 + 15.4i)T^{2} \) |
| 89 | \( 1 + (-2.53 - 15.2i)T + (-84.2 + 28.6i)T^{2} \) |
| 97 | \( 1 + (-6.09 + 7.53i)T + (-20.3 - 94.8i)T^{2} \) |
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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
−10.65442932319714107479884890314, −9.487355071993917201366428699410, −8.698422045191316289539922249008, −8.207039111721460865732260308444, −6.84763157147800633758048687282, −5.52916907296430524483392026978, −4.73007521657750157799894401335, −3.36261983895167198832315924409, −2.34315022262253750290728494954, −0.56903264891795374606393689674,
2.11637069879361549567020338768, 3.47140303092021573753283060279, 4.69519590167467681597091296977, 6.04718611496171052842888726353, 6.49459253890644687373749002685, 7.62602485829648389048484438256, 8.536520985806146910612140746538, 9.255779209490577090170533746453, 10.18326365222849336576306552073, 11.17586498200469795656764898161
