| L(s) = 1 | + (0.445 + 0.304i)2-s + (0.759 + 0.704i)3-s + (−0.624 − 1.59i)4-s + (−1.39 + 0.430i)5-s + (0.124 + 0.545i)6-s + (−2.52 − 0.795i)7-s + (0.445 − 1.95i)8-s + (−0.143 − 1.92i)9-s + (−0.754 − 0.232i)10-s + (−0.221 + 2.95i)11-s + (0.646 − 1.64i)12-s + (−0.900 + 0.433i)13-s + (−0.883 − 1.12i)14-s + (−1.36 − 0.657i)15-s + (−1.71 + 1.58i)16-s + (−1.60 + 0.242i)17-s + ⋯ |
| L(s) = 1 | + (0.315 + 0.214i)2-s + (0.438 + 0.406i)3-s + (−0.312 − 0.795i)4-s + (−0.624 + 0.192i)5-s + (0.0507 + 0.222i)6-s + (−0.953 − 0.300i)7-s + (0.157 − 0.689i)8-s + (−0.0479 − 0.640i)9-s + (−0.238 − 0.0735i)10-s + (−0.0667 + 0.890i)11-s + (0.186 − 0.475i)12-s + (−0.249 + 0.120i)13-s + (−0.236 − 0.299i)14-s + (−0.352 − 0.169i)15-s + (−0.428 + 0.397i)16-s + (−0.390 + 0.0588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.240756 - 0.552940i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.240756 - 0.552940i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.52 + 0.795i)T \) |
| 13 | \( 1 + (0.900 - 0.433i)T \) |
| good | 2 | \( 1 + (-0.445 - 0.304i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (-0.759 - 0.704i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (1.39 - 0.430i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.221 - 2.95i)T + (-10.8 - 1.63i)T^{2} \) |
| 17 | \( 1 + (1.60 - 0.242i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (3.14 + 5.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.90 + 0.738i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (2.76 + 3.46i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.69 + 2.93i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.65 + 9.31i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (1.53 - 6.72i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (1.07 + 4.73i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-7.45 - 5.08i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (1.40 + 3.58i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-14.4 - 4.45i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (0.255 - 0.650i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (6.37 - 11.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.47 - 8.12i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-6.50 + 4.43i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (2.81 + 4.87i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.7 - 5.20i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.788 + 10.5i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.975135396141800034674605158543, −9.575585270284297305475579019688, −8.752370293296028382256584741182, −7.39060937659960106290518608187, −6.66255378505378605201731909168, −5.77885811404228676051142574398, −4.25823945597099731212713576485, −4.03033350163538592280154207507, −2.47565486457241097766078824559, −0.26469963943792578366925172917,
2.25108268600673800923114580110, 3.30317694165815289322668432492, 4.07859724118526021010132319922, 5.34760442981178188555966198473, 6.49012518163259398757224894622, 7.67725250619627683980556875474, 8.242268102614329913440805895737, 8.848474611743998095784166207910, 10.05634316468520638586749953007, 11.03369250967418897598862826870
