| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 9·11-s − 12·12-s − 32·13-s − 14·14-s + 16·16-s + 114·17-s + 18·18-s − 16·19-s + 21·21-s − 18·22-s − 21·23-s − 24·24-s − 64·26-s − 27·27-s − 28·28-s − 213·29-s + 50·31-s + 32·32-s + 27·33-s + 228·34-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 − 0.246·11-s − 0.288·12-s − 0.682·13-s − 0.267·14-s + 1/4·16-s + 1.62·17-s + 0.235·18-s − 0.193·19-s + 0.218·21-s − 0.174·22-s − 0.190·23-s − 0.204·24-s − 0.482·26-s − 0.192·27-s − 0.188·28-s − 1.36·29-s + 0.289·31-s + 0.176·32-s + 0.142·33-s + 1.15·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 + 9 T + p^{3} T^{2} \) |
| 13 | \( 1 + 32 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 16 T + p^{3} T^{2} \) |
| 23 | \( 1 + 21 T + p^{3} T^{2} \) |
| 29 | \( 1 + 213 T + p^{3} T^{2} \) |
| 31 | \( 1 - 50 T + p^{3} T^{2} \) |
| 37 | \( 1 - 115 T + p^{3} T^{2} \) |
| 41 | \( 1 + 336 T + p^{3} T^{2} \) |
| 43 | \( 1 - 103 T + p^{3} T^{2} \) |
| 47 | \( 1 + 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 342 T + p^{3} T^{2} \) |
| 59 | \( 1 - 336 T + p^{3} T^{2} \) |
| 61 | \( 1 + 844 T + p^{3} T^{2} \) |
| 67 | \( 1 + 167 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1017 T + p^{3} T^{2} \) |
| 73 | \( 1 - 130 T + p^{3} T^{2} \) |
| 79 | \( 1 - 155 T + p^{3} T^{2} \) |
| 83 | \( 1 - 858 T + p^{3} T^{2} \) |
| 89 | \( 1 + 84 T + p^{3} T^{2} \) |
| 97 | \( 1 + 938 T + p^{3} T^{2} \) |
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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
−9.371567547974108754470240570697, −7.984943191398026393831476967040, −7.34936697001310401251483256215, −6.37582778421732559137196165170, −5.60076197100305433709031216877, −4.89468942395138199264624220589, −3.80008219644385509804472867776, −2.85506107217120881507740974944, −1.50769015763104944394849141099, 0,
1.50769015763104944394849141099, 2.85506107217120881507740974944, 3.80008219644385509804472867776, 4.89468942395138199264624220589, 5.60076197100305433709031216877, 6.37582778421732559137196165170, 7.34936697001310401251483256215, 7.984943191398026393831476967040, 9.371567547974108754470240570697
