L(s) = 1 | − 2-s + 2.53·3-s + 4-s + 2.74·5-s − 2.53·6-s − 8-s + 3.42·9-s − 2.74·10-s + 4.53·11-s + 2.53·12-s + 0.534·13-s + 6.95·15-s + 16-s + 17-s − 3.42·18-s + 3.53·19-s + 2.74·20-s − 4.53·22-s − 7.27·23-s − 2.53·24-s + 2.53·25-s − 0.534·26-s + 1.06·27-s − 0.955·29-s − 6.95·30-s + 5.37·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.46·3-s + 0.5·4-s + 1.22·5-s − 1.03·6-s − 0.353·8-s + 1.14·9-s − 0.867·10-s + 1.36·11-s + 0.731·12-s + 0.148·13-s + 1.79·15-s + 0.250·16-s + 0.242·17-s − 0.806·18-s + 0.810·19-s + 0.613·20-s − 0.966·22-s − 1.51·23-s − 0.517·24-s + 0.506·25-s − 0.104·26-s + 0.205·27-s − 0.177·29-s − 1.26·30-s + 0.965·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.803708422\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.803708422\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 - 2.74T + 5T^{2} \) |
| 11 | \( 1 - 4.53T + 11T^{2} \) |
| 13 | \( 1 - 0.534T + 13T^{2} \) |
| 19 | \( 1 - 3.53T + 19T^{2} \) |
| 23 | \( 1 + 7.27T + 23T^{2} \) |
| 29 | \( 1 + 0.955T + 29T^{2} \) |
| 31 | \( 1 - 5.37T + 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 + 5.84T + 43T^{2} \) |
| 47 | \( 1 + 3.37T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 - 6.53T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 9.25T + 71T^{2} \) |
| 73 | \( 1 + 4.95T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 6.64T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−9.359839623954753673436485000364, −8.665623598924040093246273663396, −8.091533076771764356188910810716, −7.09184368823974592664612312370, −6.36588694608007387003611612225, −5.46007121051940081086454540629, −3.99434380215129445507241023641, −3.16384400326111128092323105933, −2.08574113154384833703662646287, −1.44614457518201362734361588479,
1.44614457518201362734361588479, 2.08574113154384833703662646287, 3.16384400326111128092323105933, 3.99434380215129445507241023641, 5.46007121051940081086454540629, 6.36588694608007387003611612225, 7.09184368823974592664612312370, 8.091533076771764356188910810716, 8.665623598924040093246273663396, 9.359839623954753673436485000364