| L(s) = 1 | + (0.301 + 1.38i)2-s + (2.71 − 1.85i)3-s + (−1.81 + 0.832i)4-s + (1.96 − 0.147i)5-s + (3.37 + 3.19i)6-s + (−2.22 + 1.42i)7-s + (−1.69 − 2.26i)8-s + (2.85 − 7.26i)9-s + (0.794 + 2.66i)10-s + (−4.60 + 1.80i)11-s + (−3.39 + 5.62i)12-s + (0.963 + 0.768i)13-s + (−2.64 − 2.64i)14-s + (5.06 − 4.03i)15-s + (2.61 − 3.02i)16-s + (−0.724 + 0.780i)17-s + ⋯ |
| L(s) = 1 | + (0.212 + 0.977i)2-s + (1.56 − 1.06i)3-s + (−0.909 + 0.416i)4-s + (0.878 − 0.0658i)5-s + (1.37 + 1.30i)6-s + (−0.842 + 0.539i)7-s + (−0.600 − 0.799i)8-s + (0.950 − 2.42i)9-s + (0.251 + 0.844i)10-s + (−1.38 + 0.544i)11-s + (−0.980 + 1.62i)12-s + (0.267 + 0.213i)13-s + (−0.706 − 0.708i)14-s + (1.30 − 1.04i)15-s + (0.653 − 0.756i)16-s + (−0.175 + 0.189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.82783 + 0.435318i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82783 + 0.435318i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.301 - 1.38i)T \) |
| 7 | \( 1 + (2.22 - 1.42i)T \) |
| good | 3 | \( 1 + (-2.71 + 1.85i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (-1.96 + 0.147i)T + (4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (4.60 - 1.80i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.963 - 0.768i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (0.724 - 0.780i)T + (-1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.263 - 0.456i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.53 - 1.65i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.514 - 2.25i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (2.23 - 3.87i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.148 + 0.0457i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-1.00 + 2.09i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (4.03 + 8.37i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-9.44 - 1.42i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (2.10 + 0.649i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.949 + 12.6i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (0.310 + 1.00i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (0.898 + 0.518i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.2 + 2.56i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.954 - 6.33i)T + (-69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (7.88 - 4.55i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.49 - 6.89i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (8.70 + 3.41i)T + (65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 - 7.11iT - 97T^{2} \) |
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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
−12.87139432608284534923228496976, −12.47611335969163568509339897080, −9.995152475199210160865063879600, −9.219351686913492345537844382541, −8.456834435733624741333393222930, −7.42973595922272093086660545191, −6.61132768257009392082003087886, −5.47157298235666362998483699341, −3.48077222779893102072130200406, −2.25169043259422713892605854663,
2.43537699583699612841144249481, 3.21178616441235297731123680062, 4.40426382767895965407178419320, 5.69449438497478913233933814356, 7.80004486433542395916095709296, 8.861091057941341384535290462141, 9.692664511005179806175362468255, 10.25442037080331314627091650995, 10.95859461731912923303279794887, 12.96822916284343413706175962986
