| L(s) = 1 | − 2-s + 1.56·3-s + 4-s − 0.643·5-s − 1.56·6-s + 3.99·7-s − 8-s − 0.560·9-s + 0.643·10-s + 11-s + 1.56·12-s + 4.99·13-s − 3.99·14-s − 1.00·15-s + 16-s + 7.01·17-s + 0.560·18-s + 4.48·19-s − 0.643·20-s + 6.23·21-s − 22-s − 5.08·23-s − 1.56·24-s − 4.58·25-s − 4.99·26-s − 5.56·27-s + 3.99·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.901·3-s + 0.5·4-s − 0.287·5-s − 0.637·6-s + 1.50·7-s − 0.353·8-s − 0.186·9-s + 0.203·10-s + 0.301·11-s + 0.450·12-s + 1.38·13-s − 1.06·14-s − 0.259·15-s + 0.250·16-s + 1.70·17-s + 0.132·18-s + 1.02·19-s − 0.143·20-s + 1.36·21-s − 0.213·22-s − 1.05·23-s − 0.318·24-s − 0.917·25-s − 0.979·26-s − 1.07·27-s + 0.754·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.549743318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.549743318\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 + 0.643T + 5T^{2} \) |
| 7 | \( 1 - 3.99T + 7T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 - 7.01T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 + 5.08T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 - 4.63T + 37T^{2} \) |
| 41 | \( 1 - 9.16T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 4.38T + 47T^{2} \) |
| 53 | \( 1 + 3.36T + 53T^{2} \) |
| 59 | \( 1 + 0.733T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 7.54T + 83T^{2} \) |
| 89 | \( 1 + 7.76T + 89T^{2} \) |
| 97 | \( 1 - 4.28T + 97T^{2} \) |
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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
−8.400721459763416409462520702979, −7.83367180031327395629658112445, −7.46383557897549120533610227332, −6.11686380995732515056713630774, −5.61857674185916565537658456170, −4.47593159286055975051610409235, −3.60541011039628234778337725091, −2.88942309575941937468111746551, −1.72362343700478220162318324849, −1.07391880563471082859802990380,
1.07391880563471082859802990380, 1.72362343700478220162318324849, 2.88942309575941937468111746551, 3.60541011039628234778337725091, 4.47593159286055975051610409235, 5.61857674185916565537658456170, 6.11686380995732515056713630774, 7.46383557897549120533610227332, 7.83367180031327395629658112445, 8.400721459763416409462520702979
