| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 3·11-s + 2·12-s + 13-s + 16-s + 6·17-s + 18-s − 19-s + 3·22-s − 9·23-s + 2·24-s + 26-s − 4·27-s + 6·29-s + 8·31-s + 32-s + 6·33-s + 6·34-s + 36-s + 7·37-s − 38-s + 2·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.577·12-s + 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.639·22-s − 1.87·23-s + 0.408·24-s + 0.196·26-s − 0.769·27-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.04·33-s + 1.02·34-s + 1/6·36-s + 1.15·37-s − 0.162·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.570346765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.570346765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.778166366132439933784987746631, −8.083617728601138257064972002530, −7.64337135359474525755737736965, −6.39450976257199752004312106206, −6.00931489332586249061223265846, −4.76903750167706795096361582820, −3.93329655786612077672430439357, −3.27464293804874853526447821927, −2.42586044666734542265383991760, −1.32188371728262931866291570826,
1.32188371728262931866291570826, 2.42586044666734542265383991760, 3.27464293804874853526447821927, 3.93329655786612077672430439357, 4.76903750167706795096361582820, 6.00931489332586249061223265846, 6.39450976257199752004312106206, 7.64337135359474525755737736965, 8.083617728601138257064972002530, 8.778166366132439933784987746631
