| L(s) = 1 | + 2-s − 3-s + 4-s + 2.21·5-s − 6-s + 8-s + 9-s + 2.21·10-s + 3.95·11-s − 12-s + 7.06·13-s − 2.21·15-s + 16-s + 5.21·17-s + 18-s + 19-s + 2.21·20-s + 3.95·22-s + 7.23·23-s − 24-s − 0.112·25-s + 7.06·26-s − 27-s + 1.11·29-s − 2.21·30-s + 2.57·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.988·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.699·10-s + 1.19·11-s − 0.288·12-s + 1.96·13-s − 0.570·15-s + 0.250·16-s + 1.26·17-s + 0.235·18-s + 0.229·19-s + 0.494·20-s + 0.843·22-s + 1.50·23-s − 0.204·24-s − 0.0225·25-s + 1.38·26-s − 0.192·27-s + 0.206·29-s − 0.403·30-s + 0.463·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.274882268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.274882268\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2.21T + 5T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 - 7.06T + 13T^{2} \) |
| 17 | \( 1 - 5.21T + 17T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 - 1.11T + 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 + 5.85T + 37T^{2} \) |
| 41 | \( 1 - 2.67T + 41T^{2} \) |
| 43 | \( 1 + 9.76T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + 8.06T + 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 + 4.42T + 61T^{2} \) |
| 67 | \( 1 - 1.06T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 8.02T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 2.67T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
| 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
−8.144579919887432752695688078647, −7.06845017342903574025608037188, −6.43345110173128618426511835072, −6.02598001685967625240060159979, −5.36500378106148952262980672603, −4.62223479326900680077673913678, −3.59631260478483836067179617810, −3.12769954940587944187514763924, −1.53517168472584722236199304705, −1.27303734125372128739419814807,
1.27303734125372128739419814807, 1.53517168472584722236199304705, 3.12769954940587944187514763924, 3.59631260478483836067179617810, 4.62223479326900680077673913678, 5.36500378106148952262980672603, 6.02598001685967625240060159979, 6.43345110173128618426511835072, 7.06845017342903574025608037188, 8.144579919887432752695688078647
