| L(s) = 1 | − 2-s − 0.533·3-s + 4-s − 5-s + 0.533·6-s − 1.92·7-s − 8-s − 2.71·9-s + 10-s + 2.09·11-s − 0.533·12-s − 6.17·13-s + 1.92·14-s + 0.533·15-s + 16-s + 4.26·17-s + 2.71·18-s + 2.92·19-s − 20-s + 1.02·21-s − 2.09·22-s + 7.20·23-s + 0.533·24-s + 25-s + 6.17·26-s + 3.05·27-s − 1.92·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.308·3-s + 0.5·4-s − 0.447·5-s + 0.217·6-s − 0.727·7-s − 0.353·8-s − 0.905·9-s + 0.316·10-s + 0.630·11-s − 0.154·12-s − 1.71·13-s + 0.514·14-s + 0.137·15-s + 0.250·16-s + 1.03·17-s + 0.639·18-s + 0.671·19-s − 0.223·20-s + 0.224·21-s − 0.445·22-s + 1.50·23-s + 0.108·24-s + 0.200·25-s + 1.21·26-s + 0.586·27-s − 0.363·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 0.533T + 3T^{2} \) |
| 7 | \( 1 + 1.92T + 7T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 13 | \( 1 + 6.17T + 13T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 - 7.20T + 23T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 + 9.19T + 41T^{2} \) |
| 43 | \( 1 + 0.843T + 43T^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 + 2.81T + 53T^{2} \) |
| 59 | \( 1 - 1.96T + 59T^{2} \) |
| 61 | \( 1 + 2.11T + 61T^{2} \) |
| 67 | \( 1 - 6.59T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 + 4.72T + 97T^{2} \) |
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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
−7.40194265222314697493856643771, −6.92859477299959446110690889610, −6.25782648515929395413398386353, −5.26701368907264809000600582302, −4.96538878219399895664610954097, −3.49330444337484213200015101623, −3.17686641436187814526799699260, −2.19050039746017029194616536777, −0.922747953059648373534199173726, 0,
0.922747953059648373534199173726, 2.19050039746017029194616536777, 3.17686641436187814526799699260, 3.49330444337484213200015101623, 4.96538878219399895664610954097, 5.26701368907264809000600582302, 6.25782648515929395413398386353, 6.92859477299959446110690889610, 7.40194265222314697493856643771
