| L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s − 1.16·11-s − 3.16·13-s + 14-s + 16-s + 17-s + 5.16·19-s + 2·20-s − 1.16·22-s − 25-s − 3.16·26-s + 28-s + 9.48·29-s + 8.32·31-s + 32-s + 34-s + 2·35-s + 7.16·37-s + 5.16·38-s + 2·40-s − 1.16·44-s − 4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s − 0.350·11-s − 0.877·13-s + 0.267·14-s + 0.250·16-s + 0.242·17-s + 1.18·19-s + 0.447·20-s − 0.247·22-s − 0.200·25-s − 0.620·26-s + 0.188·28-s + 1.76·29-s + 1.49·31-s + 0.176·32-s + 0.171·34-s + 0.338·35-s + 1.17·37-s + 0.837·38-s + 0.316·40-s − 0.175·44-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.486955025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.486955025\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 - 7.16T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 5.16T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 2.32T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 1.67T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 - 0.324T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
| 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
−9.221605402569470019139810204373, −8.113636228720202306586346737322, −7.51421241230610793984707882429, −6.53008104792571167214076692358, −5.85688066280876825724654035434, −5.01874718867933472540518330654, −4.48167603445897652270861992348, −3.07680723954535702203805150325, −2.42859355131903772727003787368, −1.21008405321955923840890489064,
1.21008405321955923840890489064, 2.42859355131903772727003787368, 3.07680723954535702203805150325, 4.48167603445897652270861992348, 5.01874718867933472540518330654, 5.85688066280876825724654035434, 6.53008104792571167214076692358, 7.51421241230610793984707882429, 8.113636228720202306586346737322, 9.221605402569470019139810204373
