| L(s) = 1 | + 3-s − 0.467·5-s − 7-s + 9-s + 4.87·11-s + 4.56·13-s − 0.467·15-s + 6.09·17-s − 1.34·19-s − 21-s + 4.09·23-s − 4.78·25-s + 27-s − 7.78·29-s + 4.40·31-s + 4.87·33-s + 0.467·35-s − 4.40·37-s + 4.56·39-s + 6.09·41-s + 4.15·43-s − 0.467·45-s − 6.68·47-s + 49-s + 6.09·51-s − 1.34·53-s − 2.27·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.208·5-s − 0.377·7-s + 0.333·9-s + 1.47·11-s + 1.26·13-s − 0.120·15-s + 1.47·17-s − 0.308·19-s − 0.218·21-s + 0.854·23-s − 0.956·25-s + 0.192·27-s − 1.44·29-s + 0.791·31-s + 0.848·33-s + 0.0789·35-s − 0.724·37-s + 0.730·39-s + 0.951·41-s + 0.633·43-s − 0.0696·45-s − 0.975·47-s + 0.142·49-s + 0.853·51-s − 0.184·53-s − 0.307·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.873540924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.873540924\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 0.467T + 5T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 29 | \( 1 + 7.78T + 29T^{2} \) |
| 31 | \( 1 - 4.40T + 31T^{2} \) |
| 37 | \( 1 + 4.40T + 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 - 4.15T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 5.49T + 61T^{2} \) |
| 67 | \( 1 + 5.90T + 67T^{2} \) |
| 71 | \( 1 - 4.72T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 7.96T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.226282805910185184204821443244, −7.52541349362539035551979586322, −6.76248184445161341889461815991, −6.10392175853011909117872336959, −5.39410047363222688646328708383, −4.12878381928361995599966740069, −3.72738776807252615646009895441, −3.05302708842934766133231873861, −1.76630492401445512103418269703, −0.962092803943401313577110750357,
0.962092803943401313577110750357, 1.76630492401445512103418269703, 3.05302708842934766133231873861, 3.72738776807252615646009895441, 4.12878381928361995599966740069, 5.39410047363222688646328708383, 6.10392175853011909117872336959, 6.76248184445161341889461815991, 7.52541349362539035551979586322, 8.226282805910185184204821443244
