| L(s) = 1 | − 1.53·2-s + 3-s + 0.347·4-s − 0.120·5-s − 1.53·6-s + 7-s + 2.53·8-s + 9-s + 0.184·10-s − 4.57·11-s + 0.347·12-s − 1.53·14-s − 0.120·15-s − 4.57·16-s + 4.92·17-s − 1.53·18-s + 0.758·19-s − 0.0418·20-s + 21-s + 7.00·22-s − 5.47·23-s + 2.53·24-s − 4.98·25-s + 27-s + 0.347·28-s + 10.6·29-s + 0.184·30-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.577·3-s + 0.173·4-s − 0.0539·5-s − 0.625·6-s + 0.377·7-s + 0.895·8-s + 0.333·9-s + 0.0584·10-s − 1.37·11-s + 0.100·12-s − 0.409·14-s − 0.0311·15-s − 1.14·16-s + 1.19·17-s − 0.361·18-s + 0.174·19-s − 0.00936·20-s + 0.218·21-s + 1.49·22-s − 1.14·23-s + 0.516·24-s − 0.997·25-s + 0.192·27-s + 0.0656·28-s + 1.97·29-s + 0.0337·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 5 | \( 1 + 0.120T + 5T^{2} \) |
| 11 | \( 1 + 4.57T + 11T^{2} \) |
| 17 | \( 1 - 4.92T + 17T^{2} \) |
| 19 | \( 1 - 0.758T + 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 8.63T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 1.26T + 43T^{2} \) |
| 47 | \( 1 + 5.94T + 47T^{2} \) |
| 53 | \( 1 - 9.49T + 53T^{2} \) |
| 59 | \( 1 - 8.99T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 6.75T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.317370602917083581750575614482, −7.64579197743957170947440902368, −7.27582792300198877702129383545, −5.96086739752975867862990601692, −5.13722369561444119014518841348, −4.32972735085833141446965013894, −3.30863173766662464538989995581, −2.28635713664875204388941857071, −1.36021080355939734298174062422, 0,
1.36021080355939734298174062422, 2.28635713664875204388941857071, 3.30863173766662464538989995581, 4.32972735085833141446965013894, 5.13722369561444119014518841348, 5.96086739752975867862990601692, 7.27582792300198877702129383545, 7.64579197743957170947440902368, 8.317370602917083581750575614482
