| L(s) = 1 | + 2.67·2-s − 3-s + 5.15·4-s − 1.48·5-s − 2.67·6-s + 0.193·7-s + 8.44·8-s + 9-s − 3.96·10-s − 0.156·11-s − 5.15·12-s − 6.15·13-s + 0.518·14-s + 1.48·15-s + 12.2·16-s + 3.76·17-s + 2.67·18-s − 7.50·19-s − 7.63·20-s − 0.193·21-s − 0.418·22-s + 3.09·23-s − 8.44·24-s − 2.80·25-s − 16.4·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.577·3-s + 2.57·4-s − 0.662·5-s − 1.09·6-s + 0.0733·7-s + 2.98·8-s + 0.333·9-s − 1.25·10-s − 0.0471·11-s − 1.48·12-s − 1.70·13-s + 0.138·14-s + 0.382·15-s + 3.06·16-s + 0.913·17-s + 0.630·18-s − 1.72·19-s − 1.70·20-s − 0.0423·21-s − 0.0891·22-s + 0.645·23-s − 1.72·24-s − 0.561·25-s − 3.22·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.573448148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.573448148\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 - 0.193T + 7T^{2} \) |
| 11 | \( 1 + 0.156T + 11T^{2} \) |
| 13 | \( 1 + 6.15T + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 + 7.50T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 - 7.57T + 31T^{2} \) |
| 37 | \( 1 - 4.11T + 37T^{2} \) |
| 41 | \( 1 - 8.98T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 + 2.41T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 6.73T + 61T^{2} \) |
| 71 | \( 1 + 1.11T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 0.936T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 3.64T + 97T^{2} \) |
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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
−12.51082351673902225117013431844, −11.81364208370060015427757713475, −11.05369098783096015436196321786, −9.977376212892732512484683596626, −7.84251039046263156259482192993, −6.99116427451914772920585812728, −5.87906548028695731574511740212, −4.84080867293495982915882213041, −4.01205434797344911011165284834, −2.48610174301529954958031365527,
2.48610174301529954958031365527, 4.01205434797344911011165284834, 4.84080867293495982915882213041, 5.87906548028695731574511740212, 6.99116427451914772920585812728, 7.84251039046263156259482192993, 9.977376212892732512484683596626, 11.05369098783096015436196321786, 11.81364208370060015427757713475, 12.51082351673902225117013431844
