L(s) = 1 | + 2-s + 3-s + 4-s − 1.65·5-s + 6-s + 0.184·7-s + 8-s + 9-s − 1.65·10-s − 4.34·11-s + 12-s − 6.47·13-s + 0.184·14-s − 1.65·15-s + 16-s − 2.12·17-s + 18-s − 1.65·20-s + 0.184·21-s − 4.34·22-s + 0.106·23-s + 24-s − 2.26·25-s − 6.47·26-s + 27-s + 0.184·28-s + 3.98·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.739·5-s + 0.408·6-s + 0.0698·7-s + 0.353·8-s + 0.333·9-s − 0.522·10-s − 1.31·11-s + 0.288·12-s − 1.79·13-s + 0.0493·14-s − 0.426·15-s + 0.250·16-s − 0.514·17-s + 0.235·18-s − 0.369·20-s + 0.0403·21-s − 0.926·22-s + 0.0221·23-s + 0.204·24-s − 0.453·25-s − 1.26·26-s + 0.192·27-s + 0.0349·28-s + 0.740·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 - 0.184T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 23 | \( 1 - 0.106T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 + 3.22T + 31T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 + 8.63T + 41T^{2} \) |
| 43 | \( 1 - 0.0418T + 43T^{2} \) |
| 47 | \( 1 + 7.92T + 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 3.59T + 61T^{2} \) |
| 67 | \( 1 + 7.98T + 67T^{2} \) |
| 71 | \( 1 + 4.04T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 8.45T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.393966050423334239480763095022, −7.81974885616524817472191612414, −7.27565899611669356034811712442, −6.40381390401041954300857703730, −5.05947302160295411880069060114, −4.80814198104560662311475307017, −3.69036623563143435470232950365, −2.81164817411683596592111213932, −2.06718988207253053120553986752, 0,
2.06718988207253053120553986752, 2.81164817411683596592111213932, 3.69036623563143435470232950365, 4.80814198104560662311475307017, 5.05947302160295411880069060114, 6.40381390401041954300857703730, 7.27565899611669356034811712442, 7.81974885616524817472191612414, 8.393966050423334239480763095022