| L(s) = 1 | + 2.46·2-s + 4.09·4-s − 0.386·5-s + 0.376·7-s + 5.16·8-s − 0.954·10-s − 2.23·11-s − 3.26·13-s + 0.928·14-s + 4.57·16-s + 0.764·19-s − 1.58·20-s − 5.52·22-s − 5.94·23-s − 4.85·25-s − 8.06·26-s + 1.53·28-s − 1.65·29-s − 10.1·31-s + 0.947·32-s − 0.145·35-s − 3.62·37-s + 1.88·38-s − 1.99·40-s − 9.48·41-s + 5.08·43-s − 9.15·44-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 2.04·4-s − 0.172·5-s + 0.142·7-s + 1.82·8-s − 0.301·10-s − 0.674·11-s − 0.906·13-s + 0.248·14-s + 1.14·16-s + 0.175·19-s − 0.354·20-s − 1.17·22-s − 1.23·23-s − 0.970·25-s − 1.58·26-s + 0.290·28-s − 0.308·29-s − 1.82·31-s + 0.167·32-s − 0.0245·35-s − 0.595·37-s + 0.305·38-s − 0.316·40-s − 1.48·41-s + 0.775·43-s − 1.38·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 5 | \( 1 + 0.386T + 5T^{2} \) |
| 7 | \( 1 - 0.376T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 3.26T + 13T^{2} \) |
| 19 | \( 1 - 0.764T + 19T^{2} \) |
| 23 | \( 1 + 5.94T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 3.62T + 37T^{2} \) |
| 41 | \( 1 + 9.48T + 41T^{2} \) |
| 43 | \( 1 - 5.08T + 43T^{2} \) |
| 47 | \( 1 - 0.106T + 47T^{2} \) |
| 53 | \( 1 - 5.36T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 - 2.30T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 - 5.54T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−7.26208291986822374754226606573, −6.74799713922648634643762761120, −5.67543532420109661346854456056, −5.46864365710571973944433355676, −4.70897512360413227505039996198, −3.89781377382134014814038931414, −3.43318368226832562051344738792, −2.36488589944636581298519808248, −1.90068885325955846184483028307, 0,
1.90068885325955846184483028307, 2.36488589944636581298519808248, 3.43318368226832562051344738792, 3.89781377382134014814038931414, 4.70897512360413227505039996198, 5.46864365710571973944433355676, 5.67543532420109661346854456056, 6.74799713922648634643762761120, 7.26208291986822374754226606573
