| L(s) = 1 | + 2.41·2-s − 3·3-s − 2.17·4-s + 19.8·5-s − 7.24·6-s − 24.5·8-s + 9·9-s + 48.0·10-s + 23.9·11-s + 6.51·12-s + 87.3·13-s − 59.6·15-s − 41.9·16-s − 5.63·17-s + 21.7·18-s + 64.8·19-s − 43.2·20-s + 57.7·22-s − 25.5·23-s + 73.6·24-s + 270.·25-s + 210.·26-s − 27·27-s + 60.3·29-s − 144.·30-s − 122.·31-s + 95.2·32-s + ⋯ |
| L(s) = 1 | + 0.853·2-s − 0.577·3-s − 0.271·4-s + 1.77·5-s − 0.492·6-s − 1.08·8-s + 0.333·9-s + 1.51·10-s + 0.656·11-s + 0.156·12-s + 1.86·13-s − 1.02·15-s − 0.654·16-s − 0.0804·17-s + 0.284·18-s + 0.783·19-s − 0.483·20-s + 0.560·22-s − 0.232·23-s + 0.626·24-s + 2.16·25-s + 1.59·26-s − 0.192·27-s + 0.386·29-s − 0.877·30-s − 0.710·31-s + 0.526·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.570168250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.570168250\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.41T + 8T^{2} \) |
| 5 | \( 1 - 19.8T + 125T^{2} \) |
| 11 | \( 1 - 23.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.63T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 25.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 60.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 56.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 501.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 375.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 3.75T + 2.26e5T^{2} \) |
| 67 | \( 1 + 813.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 165.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 138.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 621.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 285.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 603.T + 9.12e5T^{2} \) |
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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
−12.89013581185121024513387114812, −11.77658331354819561191606962764, −10.59945077944475582313314238480, −9.532896659771571726199669550105, −8.725739071186004366609981559003, −6.50192839675392189190964801012, −5.93476750785125016658678600706, −4.98876506615176824019409913789, −3.45661462245391568881073052802, −1.45458489799164087560544367544,
1.45458489799164087560544367544, 3.45661462245391568881073052802, 4.98876506615176824019409913789, 5.93476750785125016658678600706, 6.50192839675392189190964801012, 8.725739071186004366609981559003, 9.532896659771571726199669550105, 10.59945077944475582313314238480, 11.77658331354819561191606962764, 12.89013581185121024513387114812
