| L(s) = 1 | + 0.762·2-s − 1.41·4-s + 1.72·5-s + 1.91·7-s − 2.60·8-s + 1.31·10-s + 11-s + 2.75·13-s + 1.46·14-s + 0.847·16-s − 6.48·17-s − 3.45·19-s − 2.44·20-s + 0.762·22-s + 2.31·23-s − 2.01·25-s + 2.09·26-s − 2.71·28-s − 3.30·29-s − 8.07·31-s + 5.86·32-s − 4.94·34-s + 3.31·35-s − 6.14·37-s − 2.63·38-s − 4.50·40-s − 5.53·41-s + ⋯ |
| L(s) = 1 | + 0.539·2-s − 0.709·4-s + 0.772·5-s + 0.724·7-s − 0.921·8-s + 0.416·10-s + 0.301·11-s + 0.763·13-s + 0.390·14-s + 0.211·16-s − 1.57·17-s − 0.792·19-s − 0.547·20-s + 0.162·22-s + 0.482·23-s − 0.403·25-s + 0.411·26-s − 0.513·28-s − 0.614·29-s − 1.45·31-s + 1.03·32-s − 0.848·34-s + 0.559·35-s − 1.01·37-s − 0.427·38-s − 0.712·40-s − 0.863·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.762T + 2T^{2} \) |
| 5 | \( 1 - 1.72T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 + 6.48T + 17T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 23 | \( 1 - 2.31T + 23T^{2} \) |
| 29 | \( 1 + 3.30T + 29T^{2} \) |
| 31 | \( 1 + 8.07T + 31T^{2} \) |
| 37 | \( 1 + 6.14T + 37T^{2} \) |
| 41 | \( 1 + 5.53T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 3.58T + 47T^{2} \) |
| 53 | \( 1 + 0.104T + 53T^{2} \) |
| 59 | \( 1 + 8.73T + 59T^{2} \) |
| 67 | \( 1 + 1.77T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 - 6.67T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 7.45T + 83T^{2} \) |
| 89 | \( 1 + 0.855T + 89T^{2} \) |
| 97 | \( 1 - 7.45T + 97T^{2} \) |
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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
−7.79426338724709381857752018848, −6.79873416618021383393523148992, −6.14133464295752876549530906941, −5.52534778719887176046452512810, −4.79866916561931312674552114365, −4.15334832518500899848293543130, −3.44663446345828572945787051531, −2.24946107185812047457833056016, −1.52117201510003304347618333341, 0,
1.52117201510003304347618333341, 2.24946107185812047457833056016, 3.44663446345828572945787051531, 4.15334832518500899848293543130, 4.79866916561931312674552114365, 5.52534778719887176046452512810, 6.14133464295752876549530906941, 6.79873416618021383393523148992, 7.79426338724709381857752018848
