| L(s) = 1 | + 0.696·3-s − 1.84·5-s − 0.816·7-s − 2.51·9-s + 0.991·11-s − 13-s − 1.28·15-s + 6.04·17-s + 1.01·19-s − 0.568·21-s + 2.81·23-s − 1.59·25-s − 3.84·27-s − 29-s + 5.49·31-s + 0.690·33-s + 1.50·35-s + 4.58·37-s − 0.696·39-s − 10.1·41-s − 0.776·43-s + 4.64·45-s − 8.28·47-s − 6.33·49-s + 4.20·51-s + 8.01·53-s − 1.82·55-s + ⋯ |
| L(s) = 1 | + 0.402·3-s − 0.825·5-s − 0.308·7-s − 0.838·9-s + 0.298·11-s − 0.277·13-s − 0.331·15-s + 1.46·17-s + 0.233·19-s − 0.124·21-s + 0.586·23-s − 0.319·25-s − 0.739·27-s − 0.185·29-s + 0.986·31-s + 0.120·33-s + 0.254·35-s + 0.753·37-s − 0.111·39-s − 1.58·41-s − 0.118·43-s + 0.691·45-s − 1.20·47-s − 0.904·49-s + 0.589·51-s + 1.10·53-s − 0.246·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 0.696T + 3T^{2} \) |
| 5 | \( 1 + 1.84T + 5T^{2} \) |
| 7 | \( 1 + 0.816T + 7T^{2} \) |
| 11 | \( 1 - 0.991T + 11T^{2} \) |
| 17 | \( 1 - 6.04T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 - 2.81T + 23T^{2} \) |
| 31 | \( 1 - 5.49T + 31T^{2} \) |
| 37 | \( 1 - 4.58T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 0.776T + 43T^{2} \) |
| 47 | \( 1 + 8.28T + 47T^{2} \) |
| 53 | \( 1 - 8.01T + 53T^{2} \) |
| 59 | \( 1 - 5.45T + 59T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 - 9.49T + 73T^{2} \) |
| 79 | \( 1 - 2.74T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 3.12T + 97T^{2} \) |
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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
−7.959582826833124186151757389783, −7.11347047385164698000720664872, −6.36642054632768303987390010786, −5.52925959863639434049222453134, −4.85216725422559718311891416481, −3.78520127664668088332614010001, −3.30760564402289120924557826262, −2.55711680761688534488709158408, −1.24371808111022863649447713971, 0,
1.24371808111022863649447713971, 2.55711680761688534488709158408, 3.30760564402289120924557826262, 3.78520127664668088332614010001, 4.85216725422559718311891416481, 5.52925959863639434049222453134, 6.36642054632768303987390010786, 7.11347047385164698000720664872, 7.959582826833124186151757389783
