| L(s) = 1 | − 2.49·2-s − 3-s + 4.20·4-s − 4.21·5-s + 2.49·6-s + 7-s − 5.49·8-s + 9-s + 10.5·10-s + 2.13·11-s − 4.20·12-s + 1.56·13-s − 2.49·14-s + 4.21·15-s + 5.28·16-s + 2.09·17-s − 2.49·18-s + 6.30·19-s − 17.7·20-s − 21-s − 5.31·22-s + 8.28·23-s + 5.49·24-s + 12.7·25-s − 3.89·26-s − 27-s + 4.20·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 0.577·3-s + 2.10·4-s − 1.88·5-s + 1.01·6-s + 0.377·7-s − 1.94·8-s + 0.333·9-s + 3.32·10-s + 0.643·11-s − 1.21·12-s + 0.434·13-s − 0.665·14-s + 1.08·15-s + 1.32·16-s + 0.507·17-s − 0.587·18-s + 1.44·19-s − 3.96·20-s − 0.218·21-s − 1.13·22-s + 1.72·23-s + 1.12·24-s + 2.55·25-s − 0.764·26-s − 0.192·27-s + 0.795·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6391230514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6391230514\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 227 | \( 1 + T \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 5 | \( 1 + 4.21T + 5T^{2} \) |
| 11 | \( 1 - 2.13T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 19 | \( 1 - 6.30T + 19T^{2} \) |
| 23 | \( 1 - 8.28T + 23T^{2} \) |
| 29 | \( 1 + 5.76T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 - 3.06T + 43T^{2} \) |
| 47 | \( 1 - 7.48T + 47T^{2} \) |
| 53 | \( 1 - 5.01T + 53T^{2} \) |
| 59 | \( 1 + 0.450T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 - 6.84T + 67T^{2} \) |
| 71 | \( 1 + 4.86T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 5.34T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 3.75T + 89T^{2} \) |
| 97 | \( 1 + 2.64T + 97T^{2} \) |
| 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
−8.228133017688904150173040187628, −7.55277794690401205053561045480, −7.32399071084164390928069531017, −6.55838834654421901939613420879, −5.50516850811764293977964016207, −4.49283695849675194077939889101, −3.64976564096121841837384154864, −2.73570628229518775371064699839, −1.06487318236427454891839272404, −0.805221846808288992245411469553,
0.805221846808288992245411469553, 1.06487318236427454891839272404, 2.73570628229518775371064699839, 3.64976564096121841837384154864, 4.49283695849675194077939889101, 5.50516850811764293977964016207, 6.55838834654421901939613420879, 7.32399071084164390928069531017, 7.55277794690401205053561045480, 8.228133017688904150173040187628
