L(s) = 1 | − 1.29·2-s − 3-s − 0.334·4-s + 1.33·5-s + 1.29·6-s + 7-s + 3.01·8-s + 9-s − 1.72·10-s + 4.88·11-s + 0.334·12-s − 1.29·14-s − 1.33·15-s − 3.21·16-s − 4.58·17-s − 1.29·18-s − 2.96·19-s − 0.446·20-s − 21-s − 6.30·22-s + 6.13·23-s − 3.01·24-s − 3.21·25-s − 27-s − 0.334·28-s − 7.63·29-s + 1.72·30-s + ⋯ |
L(s) = 1 | − 0.912·2-s − 0.577·3-s − 0.167·4-s + 0.596·5-s + 0.526·6-s + 0.377·7-s + 1.06·8-s + 0.333·9-s − 0.544·10-s + 1.47·11-s + 0.0965·12-s − 0.344·14-s − 0.344·15-s − 0.804·16-s − 1.11·17-s − 0.304·18-s − 0.681·19-s − 0.0998·20-s − 0.218·21-s − 1.34·22-s + 1.27·23-s − 0.614·24-s − 0.643·25-s − 0.192·27-s − 0.0632·28-s − 1.41·29-s + 0.314·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 5 | \( 1 - 1.33T + 5T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 - 6.13T + 23T^{2} \) |
| 29 | \( 1 + 7.63T + 29T^{2} \) |
| 31 | \( 1 - 5.63T + 31T^{2} \) |
| 37 | \( 1 + 9.76T + 37T^{2} \) |
| 41 | \( 1 + 3.69T + 41T^{2} \) |
| 43 | \( 1 + 9.63T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 5.50T + 67T^{2} \) |
| 71 | \( 1 - 4.88T + 71T^{2} \) |
| 73 | \( 1 - 4.45T + 73T^{2} \) |
| 79 | \( 1 + 3.54T + 79T^{2} \) |
| 83 | \( 1 + 5.27T + 83T^{2} \) |
| 89 | \( 1 + 9.94T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−8.555209474753864813457924645132, −7.41510851030435631539077093917, −6.77767821346837596328825279257, −6.14239030828510757864899563552, −5.08553817070352006769934659725, −4.49467434349701712568704723454, −3.59914451597717953899000138551, −1.94933703246615733456041821561, −1.37646112996936646364437603292, 0,
1.37646112996936646364437603292, 1.94933703246615733456041821561, 3.59914451597717953899000138551, 4.49467434349701712568704723454, 5.08553817070352006769934659725, 6.14239030828510757864899563552, 6.77767821346837596328825279257, 7.41510851030435631539077093917, 8.555209474753864813457924645132