| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 4·11-s + 12-s + 2·13-s + 14-s + 16-s + 6·17-s − 18-s + 4·19-s − 21-s + 4·22-s + 23-s − 24-s − 2·26-s + 27-s − 28-s − 2·29-s − 8·31-s − 32-s − 4·33-s − 6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.852·22-s + 0.208·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.696·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.89990465148204, −15.16993611543716, −14.76317885564382, −14.01336691785303, −13.58348679952894, −13.03039390425789, −12.32062439041649, −12.06225988656368, −11.10110577215677, −10.63937913356898, −10.20487629522950, −9.523397060305066, −9.170844807643524, −8.466046481193313, −7.894260140830243, −7.431429023873980, −7.023663821127285, −6.018728481690588, −5.526216090302711, −4.951606492602076, −3.770814363655419, −3.314280929266553, −2.734815694488328, −1.838329546353579, −1.087079782090400, 0,
1.087079782090400, 1.838329546353579, 2.734815694488328, 3.314280929266553, 3.770814363655419, 4.951606492602076, 5.526216090302711, 6.018728481690588, 7.023663821127285, 7.431429023873980, 7.894260140830243, 8.466046481193313, 9.170844807643524, 9.523397060305066, 10.20487629522950, 10.63937913356898, 11.10110577215677, 12.06225988656368, 12.32062439041649, 13.03039390425789, 13.58348679952894, 14.01336691785303, 14.76317885564382, 15.16993611543716, 15.89990465148204
