L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 4·11-s + 12-s − 13-s + 14-s + 15-s + 16-s − 18-s + 20-s − 21-s + 4·22-s − 24-s + 25-s + 26-s + 27-s − 28-s + 6·29-s − 30-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.223·20-s − 0.218·21-s + 0.852·22-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.182·30-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.38322543007343, −12.67271051358846, −12.39407011896346, −12.08972524170794, −11.13248742912895, −10.78450672098898, −10.44909098669015, −9.947663741428105, −9.337479536870232, −9.284199160831172, −8.441362744994141, −8.215362849044321, −7.507203909852175, −7.331740902613914, −6.628334439221879, −6.087318489041774, −5.667822823623839, −4.962965018968661, −4.520523089196255, −3.748113477620033, −3.108692079657515, −2.533318647585126, −2.356820652287087, −1.472775022943460, −0.8064311344002264, 0,
0.8064311344002264, 1.472775022943460, 2.356820652287087, 2.533318647585126, 3.108692079657515, 3.748113477620033, 4.520523089196255, 4.962965018968661, 5.667822823623839, 6.087318489041774, 6.628334439221879, 7.331740902613914, 7.507203909852175, 8.215362849044321, 8.441362744994141, 9.284199160831172, 9.337479536870232, 9.947663741428105, 10.44909098669015, 10.78450672098898, 11.13248742912895, 12.08972524170794, 12.39407011896346, 12.67271051358846, 13.38322543007343