L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 13.6·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 27.3·10-s − 2.24·11-s − 12·12-s + 45.7·13-s − 14·14-s + 41.0·15-s + 16·16-s + 4.13·17-s + 18·18-s + 133.·19-s − 54.6·20-s + 21·21-s − 4.49·22-s − 23·23-s − 24·24-s + 61.8·25-s + 91.4·26-s − 27·27-s − 28·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.22·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.864·10-s − 0.0616·11-s − 0.288·12-s + 0.975·13-s − 0.267·14-s + 0.705·15-s + 0.250·16-s + 0.0589·17-s + 0.235·18-s + 1.60·19-s − 0.611·20-s + 0.218·21-s − 0.0435·22-s − 0.208·23-s − 0.204·24-s + 0.495·25-s + 0.690·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
good | 5 | \( 1 + 13.6T + 125T^{2} \) |
| 11 | \( 1 + 2.24T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.13T + 4.91e3T^{2} \) |
| 19 | \( 1 - 133.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 197.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 54.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 266.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 341.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 114.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 292.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 648.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 329.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 57.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 536.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 245.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 481.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 750.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.25e3T + 9.12e5T^{2} \) |
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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
−9.280199768865867791787383303529, −8.094629512052739604491507519291, −7.45335831784123515259125260654, −6.55625697291784403896443533897, −5.68845170836781942887092469321, −4.77072708144203366688489064690, −3.77500726212173992840142800230, −3.16786511722620630307983466101, −1.36882015803993116230536992127, 0,
1.36882015803993116230536992127, 3.16786511722620630307983466101, 3.77500726212173992840142800230, 4.77072708144203366688489064690, 5.68845170836781942887092469321, 6.55625697291784403896443533897, 7.45335831784123515259125260654, 8.094629512052739604491507519291, 9.280199768865867791787383303529