| L(s) = 1 | + 2.34·2-s + 7.09·3-s − 2.49·4-s − 10.0·5-s + 16.6·6-s − 1.59·7-s − 24.6·8-s + 23.3·9-s − 23.5·10-s + 54.2·11-s − 17.7·12-s + 20.9·13-s − 3.73·14-s − 71.2·15-s − 37.8·16-s − 66.4·17-s + 54.8·18-s − 87.8·19-s + 25.0·20-s − 11.3·21-s + 127.·22-s + 106.·23-s − 174.·24-s − 24.3·25-s + 49.0·26-s − 25.6·27-s + 3.97·28-s + ⋯ |
| L(s) = 1 | + 0.829·2-s + 1.36·3-s − 0.311·4-s − 0.897·5-s + 1.13·6-s − 0.0859·7-s − 1.08·8-s + 0.866·9-s − 0.744·10-s + 1.48·11-s − 0.426·12-s + 0.445·13-s − 0.0713·14-s − 1.22·15-s − 0.590·16-s − 0.947·17-s + 0.718·18-s − 1.06·19-s + 0.279·20-s − 0.117·21-s + 1.23·22-s + 0.969·23-s − 1.48·24-s − 0.194·25-s + 0.369·26-s − 0.183·27-s + 0.0268·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 2.34T + 8T^{2} \) |
| 3 | \( 1 - 7.09T + 27T^{2} \) |
| 5 | \( 1 + 10.0T + 125T^{2} \) |
| 7 | \( 1 + 1.59T + 343T^{2} \) |
| 11 | \( 1 - 54.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 66.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 87.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 106.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 270.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 62.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 464.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 205.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 296.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 493.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 617.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 924.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 254.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 170.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 234.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 273.T + 9.12e5T^{2} \) |
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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
−8.642506583454014494727561668614, −7.965866820249055195954876099279, −6.71687639771787635562353947022, −6.30500975359807619067571101959, −4.69587658321276303442575812759, −4.28067702481598913453889591396, −3.45637839757850587671741794540, −2.91703287137278250060827656261, −1.54083760170176664150611376556, 0,
1.54083760170176664150611376556, 2.91703287137278250060827656261, 3.45637839757850587671741794540, 4.28067702481598913453889591396, 4.69587658321276303442575812759, 6.30500975359807619067571101959, 6.71687639771787635562353947022, 7.965866820249055195954876099279, 8.642506583454014494727561668614
