| L(s) = 1 | + 0.546·2-s + 3-s − 1.70·4-s − 5-s + 0.546·6-s − 7-s − 2.02·8-s + 9-s − 0.546·10-s − 11-s − 1.70·12-s − 0.568·13-s − 0.546·14-s − 15-s + 2.29·16-s + 2.70·17-s + 0.546·18-s + 6.62·19-s + 1.70·20-s − 21-s − 0.546·22-s + 8.93·23-s − 2.02·24-s + 25-s − 0.310·26-s + 27-s + 1.70·28-s + ⋯ |
| L(s) = 1 | + 0.386·2-s + 0.577·3-s − 0.850·4-s − 0.447·5-s + 0.223·6-s − 0.377·7-s − 0.714·8-s + 0.333·9-s − 0.172·10-s − 0.301·11-s − 0.491·12-s − 0.157·13-s − 0.146·14-s − 0.258·15-s + 0.574·16-s + 0.655·17-s + 0.128·18-s + 1.51·19-s + 0.380·20-s − 0.218·21-s − 0.116·22-s + 1.86·23-s − 0.412·24-s + 0.200·25-s − 0.0609·26-s + 0.192·27-s + 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.740671324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.740671324\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.546T + 2T^{2} \) |
| 13 | \( 1 + 0.568T + 13T^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 23 | \( 1 - 8.93T + 23T^{2} \) |
| 29 | \( 1 - 5.27T + 29T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 + 2.75T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 0.0520T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 - 6.36T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 - 2.26T + 73T^{2} \) |
| 79 | \( 1 + 2.56T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−9.788760164612552487018430933491, −8.909300447377749185304620101622, −8.265538531157894107758249806283, −7.42122693331536317880620756398, −6.48237852303071604337493604741, −5.19062424828374817329144499381, −4.71013548150215139994112787272, −3.36591280884191110978989785576, −3.03120596655070389442266982397, −0.955862374071830574861273941973,
0.955862374071830574861273941973, 3.03120596655070389442266982397, 3.36591280884191110978989785576, 4.71013548150215139994112787272, 5.19062424828374817329144499381, 6.48237852303071604337493604741, 7.42122693331536317880620756398, 8.265538531157894107758249806283, 8.909300447377749185304620101622, 9.788760164612552487018430933491
