Among degree 2 L-functions of (motivic) weight greater than 1, this is the one of positive rank with the smallest analytic conductor.
| L(s) = 1 | − 5·2-s − 7·3-s + 17·4-s − 7·5-s + 35·6-s − 13·7-s − 45·8-s + 22·9-s + 35·10-s − 26·11-s − 119·12-s + 13·13-s + 65·14-s + 49·15-s + 89·16-s + 77·17-s − 110·18-s − 126·19-s − 119·20-s + 91·21-s + 130·22-s − 96·23-s + 315·24-s − 76·25-s − 65·26-s + 35·27-s − 221·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 1.34·3-s + 17/8·4-s − 0.626·5-s + 2.38·6-s − 0.701·7-s − 1.98·8-s + 0.814·9-s + 1.10·10-s − 0.712·11-s − 2.86·12-s + 0.277·13-s + 1.24·14-s + 0.843·15-s + 1.39·16-s + 1.09·17-s − 1.44·18-s − 1.52·19-s − 1.33·20-s + 0.945·21-s + 1.25·22-s − 0.870·23-s + 2.67·24-s − 0.607·25-s − 0.490·26-s + 0.249·27-s − 1.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - p T \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 13 T + p^{3} T^{2} \) |
| 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 17 | \( 1 - 77 T + p^{3} T^{2} \) |
| 19 | \( 1 + 126 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 82 T + p^{3} T^{2} \) |
| 31 | \( 1 - 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 131 T + p^{3} T^{2} \) |
| 41 | \( 1 - 336 T + p^{3} T^{2} \) |
| 43 | \( 1 + 201 T + p^{3} T^{2} \) |
| 47 | \( 1 + 105 T + p^{3} T^{2} \) |
| 53 | \( 1 + 432 T + p^{3} T^{2} \) |
| 59 | \( 1 + 294 T + p^{3} T^{2} \) |
| 61 | \( 1 + 56 T + p^{3} T^{2} \) |
| 67 | \( 1 - 478 T + p^{3} T^{2} \) |
| 71 | \( 1 - 9 T + p^{3} T^{2} \) |
| 73 | \( 1 - 98 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1304 T + p^{3} T^{2} \) |
| 83 | \( 1 + 308 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1190 T + p^{3} T^{2} \) |
| 97 | \( 1 - 70 T + p^{3} T^{2} \) |
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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
−18.64002945313011709440369128241, −17.47231804263601398254082417282, −16.49731910284015820282370154310, −15.66451581817735663308079993374, −12.33586951176125151168612275176, −11.07933729517739416617187501775, −10.00504625289844338036427985935, −8.015540090380562026102644883058, −6.30516929548526949298373195082, 0,
6.30516929548526949298373195082, 8.015540090380562026102644883058, 10.00504625289844338036427985935, 11.07933729517739416617187501775, 12.33586951176125151168612275176, 15.66451581817735663308079993374, 16.49731910284015820282370154310, 17.47231804263601398254082417282, 18.64002945313011709440369128241
