L(s) = 1 | + 0.551·2-s − 3-s − 1.69·4-s + 5-s − 0.551·6-s + 4.16·7-s − 2.03·8-s + 9-s + 0.551·10-s − 3.03·11-s + 1.69·12-s + 13-s + 2.29·14-s − 15-s + 2.26·16-s − 7.33·17-s + 0.551·18-s − 2.32·19-s − 1.69·20-s − 4.16·21-s − 1.67·22-s − 4.61·23-s + 2.03·24-s + 25-s + 0.551·26-s − 27-s − 7.07·28-s + ⋯ |
L(s) = 1 | + 0.389·2-s − 0.577·3-s − 0.848·4-s + 0.447·5-s − 0.225·6-s + 1.57·7-s − 0.720·8-s + 0.333·9-s + 0.174·10-s − 0.913·11-s + 0.489·12-s + 0.277·13-s + 0.614·14-s − 0.258·15-s + 0.567·16-s − 1.77·17-s + 0.129·18-s − 0.534·19-s − 0.379·20-s − 0.909·21-s − 0.356·22-s − 0.962·23-s + 0.415·24-s + 0.200·25-s + 0.108·26-s − 0.192·27-s − 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.589052108\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.589052108\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - 0.551T + 2T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 17 | \( 1 + 7.33T + 17T^{2} \) |
| 19 | \( 1 + 2.32T + 19T^{2} \) |
| 23 | \( 1 + 4.61T + 23T^{2} \) |
| 29 | \( 1 - 1.99T + 29T^{2} \) |
| 37 | \( 1 - 4.21T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 8.03T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 5.15T + 53T^{2} \) |
| 59 | \( 1 + 5.70T + 59T^{2} \) |
| 61 | \( 1 - 6.28T + 61T^{2} \) |
| 67 | \( 1 - 8.65T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 0.108T + 73T^{2} \) |
| 79 | \( 1 - 4.32T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 0.0930T + 97T^{2} \) |
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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
−8.124534325798836787982611338315, −7.46712610188841755081479034734, −6.38572206853567293301463068200, −5.76821523998332313932802994594, −5.16301351368111134977194022405, −4.35015667522072722170137749857, −4.24472698653203443637384739978, −2.64428218018795067938565341595, −1.89152554128376833241319951277, −0.64491761800352609011637551677,
0.64491761800352609011637551677, 1.89152554128376833241319951277, 2.64428218018795067938565341595, 4.24472698653203443637384739978, 4.35015667522072722170137749857, 5.16301351368111134977194022405, 5.76821523998332313932802994594, 6.38572206853567293301463068200, 7.46712610188841755081479034734, 8.124534325798836787982611338315