| L(s) = 1 | − 2.78·2-s − 2.10·3-s + 5.74·4-s − 2.97·5-s + 5.86·6-s − 1.34·7-s − 10.4·8-s + 1.44·9-s + 8.26·10-s − 11-s − 12.1·12-s − 3.44·13-s + 3.74·14-s + 6.26·15-s + 17.5·16-s + 4.36·17-s − 4.02·18-s + 19-s − 17.0·20-s + 2.83·21-s + 2.78·22-s + 8.16·23-s + 21.9·24-s + 3.82·25-s + 9.59·26-s + 3.27·27-s − 7.73·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 1.21·3-s + 2.87·4-s − 1.32·5-s + 2.39·6-s − 0.508·7-s − 3.68·8-s + 0.482·9-s + 2.61·10-s − 0.301·11-s − 3.49·12-s − 0.955·13-s + 1.00·14-s + 1.61·15-s + 4.38·16-s + 1.05·17-s − 0.949·18-s + 0.229·19-s − 3.81·20-s + 0.618·21-s + 0.593·22-s + 1.70·23-s + 4.48·24-s + 0.765·25-s + 1.88·26-s + 0.630·27-s − 1.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1795800740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1795800740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 3 | \( 1 + 2.10T + 3T^{2} \) |
| 5 | \( 1 + 2.97T + 5T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 23 | \( 1 - 8.16T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 + 3.36T + 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 - 6.44T + 47T^{2} \) |
| 53 | \( 1 - 9.21T + 53T^{2} \) |
| 59 | \( 1 + 4.84T + 59T^{2} \) |
| 61 | \( 1 - 0.802T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 2.41T + 71T^{2} \) |
| 73 | \( 1 + 2.83T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 - 8.70T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−11.94529923937021196087687743823, −11.22098693660508106124641914392, −10.45724652512014203308362612835, −9.546410070766347647078130505933, −8.347078056621104483505827273813, −7.41259278398060849759157537186, −6.74209018503102773629951519421, −5.36806905607139323352477802686, −3.08140952855753023029935480399, −0.63699790344572476713681412121,
0.63699790344572476713681412121, 3.08140952855753023029935480399, 5.36806905607139323352477802686, 6.74209018503102773629951519421, 7.41259278398060849759157537186, 8.347078056621104483505827273813, 9.546410070766347647078130505933, 10.45724652512014203308362612835, 11.22098693660508106124641914392, 11.94529923937021196087687743823
