| L(s) = 1 | + (−0.415 − 0.909i)2-s + 0.647·3-s + (−0.654 + 0.755i)4-s + (−2.94 + 0.863i)5-s + (−0.268 − 0.588i)6-s + (−0.524 + 3.64i)7-s + (0.959 + 0.281i)8-s − 2.58·9-s + (2.00 + 2.31i)10-s + (0.347 + 3.29i)11-s + (−0.423 + 0.489i)12-s + (0.708 − 0.818i)13-s + (3.53 − 1.03i)14-s + (−1.90 + 0.559i)15-s + (−0.142 − 0.989i)16-s + (−0.910 + 0.584i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + 0.373·3-s + (−0.327 + 0.377i)4-s + (−1.31 + 0.386i)5-s + (−0.109 − 0.240i)6-s + (−0.198 + 1.37i)7-s + (0.339 + 0.0996i)8-s − 0.860·9-s + (0.634 + 0.732i)10-s + (0.104 + 0.994i)11-s + (−0.122 + 0.141i)12-s + (0.196 − 0.226i)13-s + (0.945 − 0.277i)14-s + (−0.491 + 0.144i)15-s + (−0.0355 − 0.247i)16-s + (−0.220 + 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0629 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0629 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.443746 + 0.416650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.443746 + 0.416650i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 11 | \( 1 + (-0.347 - 3.29i)T \) |
| good | 3 | \( 1 - 0.647T + 3T^{2} \) |
| 5 | \( 1 + (2.94 - 0.863i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (0.524 - 3.64i)T + (-6.71 - 1.97i)T^{2} \) |
| 13 | \( 1 + (-0.708 + 0.818i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.910 - 0.584i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.422 - 0.271i)T + (7.89 + 17.2i)T^{2} \) |
| 23 | \( 1 + (0.519 + 3.60i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 + (3.72 + 2.39i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-6.34 - 7.32i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-6.89 - 7.95i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (1.31 + 2.87i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (4.68 + 1.37i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (0.809 - 1.77i)T + (-30.7 - 35.5i)T^{2} \) |
| 53 | \( 1 + (1.11 - 7.73i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-3.02 + 6.63i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (0.587 - 1.28i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (4.65 + 10.1i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-3.07 - 1.97i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (0.163 + 1.13i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-4.89 + 1.43i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.281 + 1.96i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (6.30 - 4.05i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-13.6 - 4.01i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.96732759634162065877410774667, −11.69183193348971998458174359807, −10.53269951207408702388067394696, −9.338840781575846219392520545878, −8.484323645995284989470375040849, −7.80141948413229810749907426059, −6.37908666947132655638727244644, −4.80431335173174104141850713899, −3.41707692211962305617017247009, −2.44234133792519952855401823546,
0.51132995999349634627723243170, 3.45924750094157330946476917561, 4.32905919858560545856045350885, 5.87660459892860085557703313507, 7.18418054907084594284994956600, 7.955878996333180467922826638484, 8.643177036188703244532304945976, 9.754621324971324600121401207794, 11.13473945046584442999002608023, 11.54965975833677336427799706569
