| L(s) = 1 | − 2.90·3-s − 3.52·7-s + 5.42·9-s + 3.80·11-s + 2.62·13-s − 5.80·17-s − 5.05·19-s + 10.2·21-s − 0.474·23-s − 7.05·27-s + 2·29-s − 2.75·31-s − 11.0·33-s + 7.18·37-s − 7.61·39-s + 5.18·41-s + 1.95·43-s + 5.33·47-s + 5.42·49-s + 16.8·51-s + 5.37·53-s + 14.6·57-s + 5.05·59-s + 12.2·61-s − 19.1·63-s + 7.76·67-s + 1.37·69-s + ⋯ |
| L(s) = 1 | − 1.67·3-s − 1.33·7-s + 1.80·9-s + 1.14·11-s + 0.727·13-s − 1.40·17-s − 1.15·19-s + 2.23·21-s − 0.0989·23-s − 1.35·27-s + 0.371·29-s − 0.494·31-s − 1.92·33-s + 1.18·37-s − 1.21·39-s + 0.809·41-s + 0.297·43-s + 0.777·47-s + 0.775·49-s + 2.36·51-s + 0.738·53-s + 1.94·57-s + 0.657·59-s + 1.56·61-s − 2.41·63-s + 0.948·67-s + 0.165·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 + 5.05T + 19T^{2} \) |
| 23 | \( 1 + 0.474T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 - 1.95T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 - 5.05T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 7.76T + 67T^{2} \) |
| 71 | \( 1 + 4.85T + 71T^{2} \) |
| 73 | \( 1 - 6.66T + 73T^{2} \) |
| 79 | \( 1 + 5.24T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.425432803874883063667986072153, −6.97917109930870249771512621097, −6.66355546981959739437055428346, −6.13459672552349032301092094530, −5.50570409185369587857246815703, −4.18686997087532913995090902005, −4.03019006775387858591872711497, −2.47883121643524020694036765916, −1.07941972262805015327972476783, 0,
1.07941972262805015327972476783, 2.47883121643524020694036765916, 4.03019006775387858591872711497, 4.18686997087532913995090902005, 5.50570409185369587857246815703, 6.13459672552349032301092094530, 6.66355546981959739437055428346, 6.97917109930870249771512621097, 8.425432803874883063667986072153
