| L(s) = 1 | − 1.22·2-s + 3-s − 0.504·4-s − 1.22·6-s − 4.18·7-s + 3.06·8-s + 9-s + 1.89·11-s − 0.504·12-s + 13-s + 5.11·14-s − 2.73·16-s + 1.73·17-s − 1.22·18-s − 1.11·19-s − 4.18·21-s − 2.32·22-s − 9.30·23-s + 3.06·24-s − 1.22·26-s + 27-s + 2.10·28-s + 5.00·29-s + 10.0·31-s − 2.77·32-s + 1.89·33-s − 2.12·34-s + ⋯ |
| L(s) = 1 | − 0.864·2-s + 0.577·3-s − 0.252·4-s − 0.499·6-s − 1.58·7-s + 1.08·8-s + 0.333·9-s + 0.572·11-s − 0.145·12-s + 0.277·13-s + 1.36·14-s − 0.684·16-s + 0.421·17-s − 0.288·18-s − 0.256·19-s − 0.912·21-s − 0.494·22-s − 1.93·23-s + 0.625·24-s − 0.239·26-s + 0.192·27-s + 0.398·28-s + 0.929·29-s + 1.79·31-s − 0.490·32-s + 0.330·33-s − 0.364·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9053876492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9053876492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.22T + 2T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 - 1.89T + 11T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 + 9.30T + 23T^{2} \) |
| 29 | \( 1 - 5.00T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 8.02T + 41T^{2} \) |
| 43 | \( 1 + 1.00T + 43T^{2} \) |
| 47 | \( 1 + 5.63T + 47T^{2} \) |
| 53 | \( 1 + 0.174T + 53T^{2} \) |
| 59 | \( 1 - 8.64T + 59T^{2} \) |
| 61 | \( 1 - 3.27T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 5.37T + 71T^{2} \) |
| 73 | \( 1 - 4.01T + 73T^{2} \) |
| 79 | \( 1 + 0.613T + 79T^{2} \) |
| 83 | \( 1 - 9.08T + 83T^{2} \) |
| 89 | \( 1 + 1.46T + 89T^{2} \) |
| 97 | \( 1 + 3.85T + 97T^{2} \) |
| 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
−9.895740764290774676049477075112, −9.314530078876690004747190832346, −8.390219619968241330880554174361, −7.83543593268310044132664552944, −6.67092998369066442547178210079, −6.04545158522943968752056109090, −4.42938328148670721798420068027, −3.67254740168040415728118428476, −2.46732342000288605243950479866, −0.834274516069525070950512985959,
0.834274516069525070950512985959, 2.46732342000288605243950479866, 3.67254740168040415728118428476, 4.42938328148670721798420068027, 6.04545158522943968752056109090, 6.67092998369066442547178210079, 7.83543593268310044132664552944, 8.390219619968241330880554174361, 9.314530078876690004747190832346, 9.895740764290774676049477075112
