| L(s) = 1 | − 2-s + 6·3-s − 7·4-s + 20·5-s − 6·6-s + 7·7-s + 15·8-s + 9·9-s − 20·10-s − 60·11-s − 42·12-s − 68·13-s − 7·14-s + 120·15-s + 41·16-s − 9·18-s − 70·19-s − 140·20-s + 42·21-s + 60·22-s + 176·23-s + 90·24-s + 275·25-s + 68·26-s − 108·27-s − 49·28-s + 90·29-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.15·3-s − 7/8·4-s + 1.78·5-s − 0.408·6-s + 0.377·7-s + 0.662·8-s + 1/3·9-s − 0.632·10-s − 1.64·11-s − 1.01·12-s − 1.45·13-s − 0.133·14-s + 2.06·15-s + 0.640·16-s − 0.117·18-s − 0.845·19-s − 1.56·20-s + 0.436·21-s + 0.581·22-s + 1.59·23-s + 0.765·24-s + 11/5·25-s + 0.512·26-s − 0.769·27-s − 0.330·28-s + 0.576·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - p T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 5 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 68 T + p^{3} T^{2} \) |
| 19 | \( 1 + 70 T + p^{3} T^{2} \) |
| 23 | \( 1 - 176 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 22 T + p^{3} T^{2} \) |
| 41 | \( 1 - 138 T + p^{3} T^{2} \) |
| 43 | \( 1 - 328 T + p^{3} T^{2} \) |
| 47 | \( 1 + 12 T + p^{3} T^{2} \) |
| 53 | \( 1 + 234 T + p^{3} T^{2} \) |
| 59 | \( 1 + 54 T + p^{3} T^{2} \) |
| 61 | \( 1 + 44 T + p^{3} T^{2} \) |
| 67 | \( 1 + 596 T + p^{3} T^{2} \) |
| 71 | \( 1 + 200 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1122 T + p^{3} T^{2} \) |
| 79 | \( 1 + 480 T + p^{3} T^{2} \) |
| 83 | \( 1 + 838 T + p^{3} T^{2} \) |
| 89 | \( 1 - 778 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1142 T + p^{3} T^{2} \) |
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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
−8.672860537990974570564801969137, −7.77461277858148526791521862739, −7.17303226021949612561219220912, −5.75587429591641957354222486764, −5.18245069940158503532695466139, −4.49240016796289910963663203287, −2.87808099694703156613029811080, −2.47956246608874700658919385076, −1.50417645399165074806134334412, 0,
1.50417645399165074806134334412, 2.47956246608874700658919385076, 2.87808099694703156613029811080, 4.49240016796289910963663203287, 5.18245069940158503532695466139, 5.75587429591641957354222486764, 7.17303226021949612561219220912, 7.77461277858148526791521862739, 8.672860537990974570564801969137
